from sklearn.linear_model import LinearRegression
import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split, cross_val_predict, cross_val_score, GridSearchCV, ParameterGrid, KFold, \
RandomizedSearchCV
from sklearn.preprocessing import StandardScaler, PolynomialFeatures
from sklearn.metrics import accuracy_score, mean_absolute_error, mean_squared_error, r2_score, \
roc_curve, auc, precision_score, recall_score, confusion_matrix
import statsmodels.formula.api as smf
import statsmodels.api as sm
from patsy import dmatrix
from pyearth import Earth
from sklearn.tree import DecisionTreeClassifier, DecisionTreeRegressor
from sklearn.ensemble import BaggingRegressor,BaggingClassifier,RandomForestRegressor,RandomForestClassifier
from sklearn.neighbors import KNeighborsRegressor
import warnings
warnings.filterwarnings('ignore')
from sklearn.preprocessing import LabelEncoder
import random
from sklearn.ensemble import VotingRegressor, StackingRegressorFinal Project Code - Used Cars
data_sci
data = pd.read_csv('us_used_cars_final.csv')
data.head(2)| vin | back_legroom | body_type | city | city_fuel_economy | daysonmarket | dealer_zip | description | engine_cylinders | engine_displacement | ... | torque | transmission | transmission_display | trimId | trim_name | wheel_system | wheel_system_display | wheelbase | width | year | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 5LMCJ2C97HUL71203 | 36.8 in | SUV / Crossover | Devine | 21.0 | 40 | 78016 | *We have 19 sources of financing and second ch... | I4 | 2000.0 | ... | 270 lb-ft @ 3,000 RPM | A | Automatic | t66989 | Select FWD | FWD | Front-Wheel Drive | 105.9 in | 84.1 in | 2017 |
| 1 | 1FM5K8D87HGE07214 | 39.5 in | SUV / Crossover | Buffalo | 16.0 | 9 | 55313 | We are open for business. Purchase your nex... | V6 | 3500.0 | ... | 255 lb-ft @ 4,000 RPM | A | 6-Speed Automatic | t67141 | XLT AWD | AWD | All-Wheel Drive | 112.8 in | 90.2 in | 2017 |
2 rows × 57 columns
1 Data quality check / cleaning / preparation
Put code with comments. The comments should explain the code such that it can be easily understood. You may put text (in a markdown cell) before a large chunk of code to explain the overall purpose of the code, if it is not intuitive. Put the name of the person / persons who contributed to each code chunk / set of code chunks. An example is given below.
1.1 Distribution of response
By Hiba Khatib
#...Plot for distribution of response...#
plt.figure(figsize=(10,6))
sns.distplot(data.price)
plt.title('Price')
plt.legend()
plt.xlabel('Price')
plt.ylabel('Frequency')
plt.ticklabel_format(style='plain', axis='x')
plt.show()
# Mean and standard deviation of response #
print("Mean of price: ", np.mean(data.price))
print("Standard deviation: ", np.std(data.price))No artists with labels found to put in legend. Note that artists whose label start with an underscore are ignored when legend() is called with no argument.
Mean of price: 24568.3835
Standard deviation: 16742.99685916766
The data needs a log transformation on price.
plt.figure(figsize=(10,6))
sns.distplot(np.log(data.price))
plt.title('Price')
plt.legend()
plt.xlabel('log(Price)')
plt.ylabel('Frequency')
plt.ticklabel_format(style='plain', axis='x')
plt.show()No artists with labels found to put in legend. Note that artists whose label start with an underscore are ignored when legend() is called with no argument.
The distribution is much more uniform now with the log transformation.
# checking for outlying observations
sns.boxplot(data.price, orient = "v")<AxesSubplot:xlabel='price'>
data.price.describe()count 6000.000000
mean 24568.383500
std 16744.392283
min 1680.000000
25% 16500.000000
50% 21924.000000
75% 29995.000000
max 859000.000000
Name: price, dtype: float64
There appears to be many outliers that we can drop before performing any feature selection. Dropping the outliers should help us avoid possible issues with overfitting to the training data and the complexities that are related to outlying observations.
data_subset = data[data.price <= 30000]sns.boxplot(data_subset.price)<AxesSubplot:xlabel='price'>
#...Plot for distribution of response after removing outliers...#
plt.figure(figsize=(10,6))
sns.distplot(data_subset.price)
plt.title('Price')
plt.legend()
plt.xlabel('Price')
plt.ylabel('Frequency')
plt.ticklabel_format(style='plain', axis='x')
plt.show()
# Mean and standard deviation of response #
print("Mean of price: ", np.mean(data_subset.price))
print("Standard deviation: ", np.std(data_subset.price))No artists with labels found to put in legend. Note that artists whose label start with an underscore are ignored when legend() is called with no argument.
Mean of price: 19205.211827007945
Standard deviation: 5978.306973568847
The mean and standard deviation have significantly decreased after removing all of the outliers and now we don’t need to transform the predictor. The mean and standard deviation have also decreased significantly.
data = data_subset.copy()print("The final shape of the data after removing outliers is", data.shape)The final shape of the data after removing outliers is (4532, 57)
1.2 Data Cleaning, Preparation, and EDA
By Hiba Khatib & Emily Leibfritz
After creating all 8 base models, we decided to subset the data into two different ways to improve the analysis of our data. The first subset is the resulting features after our EDA and data preparation. The second way we subsetted the data was based on the features selected by tree based models which is summarized later in the report (done by Hiba) Our methods are described in detail below.
- Our EDA Selected Features (written and performed by Hiba & Emily)
We performed data cleaning, preparation and EDA on the data to facilitate feature selection and model training. The first thing we did was fix the data types of some of the columns. All of the boolean columns were encoded as True or False so we mapped those to 1s and 0s. Then, we found that several columns such as horsepower and torque were numerical but contained units. We removed all of the units from these columns and converted them back to numeric. Then, we dropped a set of unique identifiers which were not sigificant to our analysis. We also dropped three columns which represented the interior, and exterior color of the car. These columns had different names for the same color and we did not see them as significant predictors to engineer and encode differently. Then, we dropped a constant predictor which contained the same value for all observations as constant predictors don’t provide any meaningful insights. The final step in the cleaning and preparation of the numerical data, we dropped highly correlated features. These were features which represented the same information in two different ways such as the official name of a trim and the id associated with that name. This resulted in us keeping only 14 of the numerical predictors are subsetting for predictors with a correlation of |-.2|.
For the categorical columns, each was observed individually in order to find the best way to handle them, as just converting them all into dummies resulted in a huge dataset.
Some columns with few unique values that showed a connection to the target variable were just converted into dummies, like body_type and transmission_display.
Some columns, like engine_cylinders and transmission display had some values very common while others only had under 100 observations. These were grouped as others, and the columns were converted into dummies.
Some columns like franchise_make and model_name had a too large amount of unique values to convert into dummies, so the observations were grouped by mean value of target variable, and assigned a number between 1-6.
Some columns did not show any connection to the target variable, while often having a big amount of unique values, like the columns concerning the color of the vehicles, city or major_options. These columns were dropped.
Some columns contained the same information given by other columns already, like the pairs trim&trimID, make_name&franchise_make, engine_type&engine_cylinders where each trim, engine_type and make_name were dropped.
# feature engineering
# encoding True/False as 1/0s
bool_cols = data.dtypes[data.dtypes == bool]
bool_cols = pd.DataFrame(bool_cols).reset_index()
bool_cols = bool_cols['index']
bool_cols = list(bool_cols)
print(bool_cols)
for col in bool_cols:
data[col] = data[col].map({False: 0, True: 1})
# numerically encoding categorical variables with at most 5 unique values
cols_map = []
for c in data.columns:
col_len = (len(data[c].unique()))
if col_len <=5:
cols_map.append(c)
[cols_map.remove(c) for c in bool_cols] # removing the boolean columns from the list
# performing the encoding
le = LabelEncoder()
encoder_dict = {key: None for key in cols_map}
for c in cols_map:
encoder_dict[c] = data[c].unique()
col_name = str(c)+"_cat"
data[col_name] = le.fit_transform(data[c])
encoder_dict[col_name] = le.fit_transform(encoder_dict[c])
# removing units from numerical columns
data['front_legroom'] = data['front_legroom'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['width'] = data['width'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['height'] = data['height'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['length'] = data['length'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['wheelbase'] = data['wheelbase'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['back_legroom'] = data['back_legroom'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['maximum_seating'] = data['maximum_seating'].map(lambda x: x.lstrip('+-').rstrip('seating'))
data['fuel_tank_volume'] = data['fuel_tank_volume'].map(lambda x: x.lstrip('+-').rstrip('gal'))
data['power'] = data['power'].str[:3] # keeping only numerical value
data['torque'] = data['torque'].str[:3]
# converting data types to numerical
data.power = pd.to_numeric(data.power)
data.back_legroom = pd.to_numeric(data.back_legroom)
data.front_legroom = pd.to_numeric(data.front_legroom)
data.width = pd.to_numeric(data.width)
data.height = pd.to_numeric(data.height)
data.length = pd.to_numeric(data.length)
data.wheelbase = pd.to_numeric(data.wheelbase)
data.maximum_seating = pd.to_numeric(data.maximum_seating)
data.fuel_tank_volume = pd.to_numeric(data.fuel_tank_volume)
data.torque = pd.to_numeric(data.torque)['fleet', 'frame_damaged', 'franchise_dealer', 'has_accidents', 'isCab', 'is_new', 'salvage', 'theft_title']
data.columnsIndex(['vin', 'back_legroom', 'body_type', 'city', 'city_fuel_economy',
'daysonmarket', 'dealer_zip', 'description', 'engine_cylinders',
'engine_displacement', 'engine_type', 'exterior_color', 'fleet',
'frame_damaged', 'franchise_dealer', 'franchise_make', 'front_legroom',
'fuel_tank_volume', 'fuel_type', 'has_accidents', 'height',
'highway_fuel_economy', 'horsepower', 'interior_color', 'isCab',
'is_new', 'latitude', 'length', 'listed_date', 'listing_color',
'listing_id', 'longitude', 'main_picture_url', 'major_options',
'make_name', 'maximum_seating', 'mileage', 'model_name', 'owner_count',
'power', 'price', 'salvage', 'savings_amount', 'seller_rating', 'sp_id',
'sp_name', 'theft_title', 'torque', 'transmission',
'transmission_display', 'trimId', 'trim_name', 'wheel_system',
'wheel_system_display', 'wheelbase', 'width', 'year', 'fuel_type_cat',
'transmission_cat', 'wheel_system_cat', 'wheel_system_display_cat'],
dtype='object')
# dropping unique features
data.drop(columns = ['vin', 'listing_id', 'main_picture_url', 'description', 'sp_name', 'make_name', 'city'], inplace = True)
# color columns
data.drop(columns = ['listing_color', 'interior_color', 'exterior_color'], inplace = True)
print("Data shape after dropping unique features: ", data.shape)
# dropping constant predictor
print(data.nunique() == 1)
data.drop(columns = ['franchise_dealer'], inplace = True)
# dropping highly correlated columns
data.drop(columns = ['trim_name', 'engine_type', 'major_options', 'horsepower'], inplace = True)Data shape after dropping unique features: (4532, 51)
back_legroom False
body_type False
city_fuel_economy False
daysonmarket False
dealer_zip False
engine_cylinders False
engine_displacement False
engine_type False
fleet False
frame_damaged False
franchise_dealer True
franchise_make False
front_legroom False
fuel_tank_volume False
fuel_type False
has_accidents False
height False
highway_fuel_economy False
horsepower False
isCab False
is_new False
latitude False
length False
listed_date False
longitude False
major_options False
maximum_seating False
mileage False
model_name False
owner_count False
power False
price False
salvage False
savings_amount False
seller_rating False
sp_id False
theft_title False
torque False
transmission False
transmission_display False
trimId False
trim_name False
wheel_system False
wheel_system_display False
wheelbase False
width False
year False
fuel_type_cat False
transmission_cat False
wheel_system_cat False
wheel_system_display_cat False
dtype: bool
print("Final data shape after cleaning and dropping unnecessary features: ", data.shape)Final data shape after cleaning and dropping unnecessary features: (4532, 46)
1.3 Exploring Trends in Categorical Predictors
data_2 = data.copy()
# make seperate df to just look at categorical
o_data = data.drop(columns = ["back_legroom", "wheel_system_display", "wheelbase", "width", "front_legroom",
"fuel_tank_volume", "height", "torque", "maximum_seating","transmission_cat",
"power","city_fuel_economy","daysonmarket","dealer_zip","transmission","wheel_system_display_cat",
"fuel_type_cat","wheel_system","wheel_system_cat","fuel_type",
"engine_displacement","fleet","frame_damaged","has_accidents","highway_fuel_economy",
"isCab","is_new","latitude","longitude","mileage","owner_count","salvage",
"savings_amount","seller_rating","sp_id","theft_title","year"], axis = 1)
# taken out: "franchise_dealer"
cols = o_data.columns
name = []
unique_value = []
for i in range(0,len(cols)):
c = cols[i]
unique = len(data[c].unique())
unique_value.append(unique)
name.append(c)# Create the pandas DataFrame with object columns and their unique values
data_ = {'column': name, 'unique_values':unique_value}
o_unique_values_count = pd.DataFrame(data_)
o_unique_values_count.sort_values(by=['unique_values'])| column | unique_values | |
|---|---|---|
| 0 | body_type | 9 |
| 1 | engine_cylinders | 16 |
| 7 | transmission_display | 18 |
| 2 | franchise_make | 34 |
| 4 | listed_date | 286 |
| 5 | model_name | 289 |
| 3 | length | 333 |
| 8 | trimId | 2344 |
| 6 | price | 2485 |
From this information, we can simply visualize with boxplot any variables until make_name, to see their relevance and adjust them. After that we will look more thoroughly, maybe group data. Consider dropping anything with other 1000 unique values as we only have 6000 rows, so might not be good predictors, but will decide on case by case basis.
plt.figure(figsize=(10,4))
sns.boxplot(x="body_type",y="price", data = data)
plt.xticks(rotation=90)
plt.ylim(0,80000)
plt.plot()[]
body_type seems to have a significant trend right away. We cancheck value counts to see if some types are more common than others
data.body_type.value_counts()SUV / Crossover 2295
Sedan 1503
Pickup Truck 190
Minivan 184
Wagon 115
Hatchback 108
Coupe 98
Convertible 27
Van 12
Name: body_type, dtype: int64
Have enough observations for each value to be valid, none of the smaller groups stand out substantially, convertible is a little bit spread but that should be fine -> convert to dummies
# change spelling so colummns will be callable
data_2['body_type'] = o_data['body_type'].apply(lambda x: 'SUV_Crossover' if x == 'SUV / Crossover' else
'Pickup_Truck' if x == 'Pickup Truck' else
x)
# get dummies
data_2 = pd.get_dummies(data_2, columns=['body_type'], drop_first=True)plt.figure(figsize=(10,4))
sns.boxplot(x="engine_cylinders",y="price", data = o_data)
plt.xticks(rotation=90)
plt.ylim(0,80000)
plt.plot()[]
o_data.engine_cylinders.value_counts()I4 2776
V6 1223
V6 Flex Fuel Vehicle 156
H4 145
V8 105
V8 Flex Fuel Vehicle 45
I4 Flex Fuel Vehicle 22
I4 Diesel 16
I3 13
I6 12
H6 4
V6 Diesel 4
I5 4
V6 Biodiesel 3
I4 Hybrid 3
H4 Hybrid 1
Name: engine_cylinders, dtype: int64
Turn anything under 100 observations into ‘other’, if there is only one observation this is not useful data.
data_2['engine_cylinders'] = o_data['engine_cylinders'].apply(lambda x: 'I4' if x == 'I4' else
'V6' if x == 'V6' else
'V8' if x == 'V8' else
'V6_Flex_Fuel_Vehicle' if x == 'V6 Flex Fuel Vehicle' else
'H4' if x == 'H4' else
'other')
data_2.engine_cylinders.value_counts()I4 2776
V6 1223
V6_Flex_Fuel_Vehicle 156
H4 145
other 127
V8 105
Name: engine_cylinders, dtype: int64
plt.figure(figsize=(8,4))
sns.boxplot(x="engine_cylinders",y="price", data = data_2)
plt.xticks(rotation=90)
plt.ylim(0,80000)
plt.plot()[]
Other is naturally with bigger range, but this looks better, get dummies for it:
data_2 = pd.get_dummies(data_2, columns=['engine_cylinders'], drop_first=True)plt.figure(figsize=(10,4))
sns.boxplot(x="transmission_display",y="price", data = o_data)
plt.xticks(rotation=90)
plt.ylim(0,80000)
plt.plot()[]
Mean and range seem both to be varying, might be valuable column to predict price, check value counts:
data.transmission_display.value_counts().head()Automatic 1583
6-Speed Automatic 1157
Continuously Variable Transmission 934
8-Speed Automatic 334
9-Speed Automatic 189
Name: transmission_display, dtype: int64
data_2['transmission_display'] = o_data['transmission_display'].apply(lambda x: 'Automatic' if x == 'Automatic' else
'6_Speed_Automatic' if x == '6-Speed Automatic' else
'Continuously_Variable_Transmission' if x == 'Continuously Variable Transmission' else
'8_Speed_Automatic' if x == '8-Speed Automatic' else
'9_Speed_Automatic' if x == '9-Speed Automatic' else
'6_Speed_Automatic' if x == '6-Speed Automatic' else
'5_Speed_Automatic' if x == '5-Speed Automatic' else
'other')
plt.figure(figsize=(6,4))
sns.boxplot(x="transmission_display",y="price", data = data_2)
plt.xticks(rotation=90)
plt.ylim(0,80000)
plt.plot()[]
Looks relevant, change in mean and besides 2 relatively small range -> convert into dummies
# turn those columns into dummies
data_2 = pd.get_dummies(data_2, columns=['transmission_display'], drop_first=True)plt.figure(figsize=(12,5))
sns.boxplot(x="franchise_make",y="price", data = o_data)
plt.xticks(rotation=90)
plt.ylim(0,80000)
plt.plot()[]
This seems to be too many columns to just turn into dummies, and as brand is strongly connected to price this seems an important column to keep -> divide brands into groups depending on their mean price, and label encode these groups.
# look at mean price by group
brand_mean_price=data.groupby('franchise_make', as_index=False)['price'].mean()
brand_mean_price=brand_mean_price.sort_values(by=['price'])
brand_mean_price.head()| franchise_make | price | |
|---|---|---|
| 16 | Kia | 17610.859551 |
| 25 | Mitsubishi | 17617.980000 |
| 29 | Scion | 17778.894737 |
| 32 | Volkswagen | 18031.586957 |
| 13 | Hyundai | 18046.270270 |
# sort the brands into groups with similar means
group_0 = brand_mean_price.iloc[0:8,0]
group_1 = brand_mean_price.iloc[8:14,0]
group_2 = brand_mean_price.iloc[14:19,0]
group_3 = brand_mean_price.iloc[19:25,0]
group_4 = brand_mean_price.iloc[25:34,0]
group_5 = brand_mean_price.iloc[34:38,0]
group_6 = brand_mean_price.iloc[38:40,0]
# write function to get names
def get_names (group):
final = []
for i in range(0,len(group)):
name = group.iloc[i]
final.append(name)
return(final)
# list of names in each group
names_0 = get_names(group_0)
names_1 = get_names(group_1)
names_2 = get_names(group_2)
names_3 = get_names(group_3)
names_4 = get_names(group_4)
names_5 = get_names(group_5)
names_6 = get_names(group_6)
# replace each object with a number of their group
data_2['franchise_make'] = o_data['franchise_make'].apply(lambda x: 0 if x in names_0 else
1 if x in names_1 else
2 if x in names_2 else
3 if x in names_3 else
4 if x in names_4 else
5 if x in names_5 else
6 if x in names_6 else
'wrong')data.listed_date.value_counts().head()2020-09-02 163
2020-08-28 119
2020-09-03 110
2020-08-27 109
2020-08-30 105
Name: listed_date, dtype: int64
#simply turn into numerical
import datetime
data_2['listed_date']=pd.to_datetime(data_2["listed_date"]).dt.strftime("%Y%m%d")
data_2['listed_date']=data_2.listed_date.astype(int)
# get correlation
data_2['listed_date'].corr(data_2['price'])-0.027944370408355408
Has a negative correlation between listing date and price, rather small
data.model_name.value_counts().head()Escape 166
Rogue 146
Accord 125
Corolla 114
RAV4 108
Name: model_name, dtype: int64
Looking at both head and tail, there is a good distribution of the different model names, not one that is prelevant wherease the rest only has 1 or 2 entries -> group and turn into numeric
# look at grouped mean
make_mean_price=data.groupby('model_name', as_index=False)['price'].mean()
make_mean_price=make_mean_price.sort_values(by=['price'])
make_mean_price.head()| model_name | price | |
|---|---|---|
| 195 | QX4 | 2000.0 |
| 164 | Mariner Hybrid | 2953.0 |
| 95 | Explorer Sport Trac | 2995.0 |
| 101 | Five Hundred | 3000.0 |
| 182 | Park Avenue | 3000.0 |
# like with other column, divide 361 rows into 7 groups, different sizes but around same mean
group_0 = make_mean_price.iloc[0:100,0] # 2000-15000
group_1 = make_mean_price.iloc[100:194,0] # 15000-25000
group_2 = make_mean_price.iloc[194:237,0] # 25000-30000
group_3 = make_mean_price.iloc[237:279,0] # 30000-35000
group_4 = make_mean_price.iloc[279:329,0] # 35000-50000
group_5 = make_mean_price.iloc[319:352,0] # 50000-10000
group_6 = make_mean_price.iloc[352:362,0] # rest
# list of names in each group
names_0 = get_names(group_0)
names_1 = get_names(group_1)
names_2 = get_names(group_2)
names_3 = get_names(group_3)
names_4 = get_names(group_4)
names_5 = get_names(group_5)
names_6 = get_names(group_6)
# replace each object with a number of their group
data_2['model_name'] = o_data['model_name'].apply(lambda x: 0 if x in names_0 else
1 if x in names_1 else
2 if x in names_2 else
3 if x in names_3 else
4 if x in names_4 else
5 if x in names_5 else
6 if x in names_6 else
'wrong')# drop first letter of each row
data_2['trimId'] = data['trimId'].str.slice(1)
# convert to integer
data_2['trimId']=data_2.trimId.astype(int)
# look at correlation
data_2['trimId'].corr(data_2['price'])0.5706133697377521
# drop categorical version of columns we have numerical copy of
data_2 = data_2.drop(columns=["transmission","wheel_system_display","fuel_type","wheel_system"], axis = 1)dat_types = pd.DataFrame(data_2.dtypes)dat_types.value_counts()int64 21
uint8 19
float64 18
dtype: int64
data_2.shape(4532, 58)
data = data_2data| back_legroom | city_fuel_economy | daysonmarket | dealer_zip | engine_displacement | fleet | frame_damaged | franchise_make | front_legroom | fuel_tank_volume | ... | engine_cylinders_V6 | engine_cylinders_V6_Flex_Fuel_Vehicle | engine_cylinders_V8 | engine_cylinders_other | transmission_display_6_Speed_Automatic | transmission_display_8_Speed_Automatic | transmission_display_9_Speed_Automatic | transmission_display_Automatic | transmission_display_Continuously_Variable_Transmission | transmission_display_other | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 36.8 | 21.0 | 40 | 78016 | 2000.0 | 0 | 0 | 1 | 42.8 | 15.5 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 2 | 36.8 | 23.0 | 30 | 37129 | 2500.0 | 0 | 0 | 1 | 43.1 | 15.4 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 5 | 38.2 | 21.0 | 161 | 73069 | 2000.0 | 0 | 0 | 0 | 41.5 | 16.4 | ... | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40.3 | 15.0 | 14 | 48836 | 5700.0 | 0 | 0 | 4 | 41.0 | 26.0 | ... | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 7 | 40.9 | 21.0 | 49 | 12701 | 2400.0 | 1 | 0 | 0 | 44.1 | 18.8 | ... | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
| 5994 | 33.4 | 23.0 | 1 | 95407 | 2000.0 | 0 | 0 | 1 | 43.1 | 12.4 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 5995 | 37.0 | 20.0 | 88 | 15146 | 2000.0 | 0 | 0 | 3 | 44.3 | 18.0 | ... | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 5996 | 39.8 | 22.0 | 38 | 77065 | 2500.0 | 0 | 0 | 0 | 45.8 | 18.5 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 5997 | 37.3 | 22.0 | 92 | 84341 | 2000.0 | 0 | 0 | 1 | 43.1 | 15.7 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 5999 | 35.7 | 30.0 | 64 | 95661 | 2000.0 | 1 | 0 | 0 | 42.2 | 14.0 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
4532 rows × 58 columns
1.4 Feature Selection
final_corrs = pd.DataFrame(data.corrwith(data.price)).reset_index().sort_values(by = 0, ascending = False)
final_corr_cols = list(final_corrs.loc[(final_corrs[0] >= 0.2) | (final_corrs[0] <= -0.2)][1:]['index'])
final_corrs.loc[(final_corrs[0] >= 0.2) | (final_corrs[0] <= -0.2)][1:]| index | 0 | |
|---|---|---|
| 21 | model_name | 0.581373 |
| 34 | year | 0.574644 |
| 31 | trimId | 0.570613 |
| 23 | power | 0.406592 |
| 30 | torque | 0.386623 |
| 33 | width | 0.303995 |
| 53 | transmission_display_8_Speed_Automatic | 0.268433 |
| 32 | wheelbase | 0.256636 |
| 11 | height | 0.245438 |
| 48 | engine_cylinders_V6 | 0.245117 |
| 16 | length | 0.238048 |
| 9 | fuel_tank_volume | 0.225640 |
| 4 | engine_displacement | 0.220415 |
| 26 | savings_amount | 0.203765 |
| 7 | franchise_make | 0.202163 |
| 1 | city_fuel_economy | -0.203095 |
| 37 | wheel_system_cat | -0.203731 |
| 38 | wheel_system_display_cat | -0.208184 |
| 12 | highway_fuel_economy | -0.230709 |
| 44 | body_type_Sedan | -0.231848 |
| 47 | engine_cylinders_I4 | -0.232696 |
| 22 | owner_count | -0.383574 |
| 20 | mileage | -0.566824 |
X = data.copy()
for col in X:
if col not in final_corr_cols:
X.drop(col, axis = 1, inplace = True)
X.shape(4532, 23)
We have 23 predictors which have a reasonable correlation with price which we will be the only predictors we use in our model building.
When we develop our models, we will print the feature importances and note which features are deemed significant by each model.
# checking for any new NAs
X.isna().sum()city_fuel_economy 0
engine_displacement 0
franchise_make 0
fuel_tank_volume 1
height 1
highway_fuel_economy 0
length 1
mileage 0
model_name 0
owner_count 0
power 0
savings_amount 0
torque 0
trimId 0
wheelbase 1
width 1
year 0
wheel_system_cat 0
wheel_system_display_cat 0
body_type_Sedan 0
engine_cylinders_I4 0
engine_cylinders_V6 0
transmission_display_8_Speed_Automatic 0
dtype: int64
y = data.price
y.isna().sum()0
The original dataset had ‘–’ for this one observation. When we performed the data cleaning and prep, these observations were not considered missing but became missing after our feature engineering. We can drop this observation since it is only one row with data missing from the original source.
X.isna().sum()
X.iloc[29]
X.drop(29, inplace = True)
y.drop(29, inplace = True)X.to_csv('X.csv')
y.to_csv('y.csv')clean_X = X.copy()
clean_y = y.copy()X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state = 4)1.5 Features Selection using Tree Based Models
Using the tree based methods such as RandomForest and DecisionTrees will allow us to double check if our feature selection has been significant or if it produces better results for base models.
The original shape of our data after removing the outliers is 4532, 58.
After our data cleaning and dropping of the units, we created some NaN values which can be dropped. We can confidently drop these observations because they were initially missing but encoded as ‘–’ which is why we did not initially discover that they were missing. These are 31 observations which we can drop which brings our data shape to 4301, 58
data.shape(4532, 58)
data_copy = data.copy()
data_copy.dropna(inplace = True)
data_copy.shape(4501, 58)
X_tree = data_copy.drop(columns = ['price'])
y_tree = data_copy.price1.5.1 Base Random Forest Feature Selection
rf_features = RandomForestRegressor(random_state = 4).fit(X_tree, y_tree)
rf_feats = pd.DataFrame(zip(rf_features.feature_names_in_, rf_features.feature_importances_))rf_feats.sort_values(by = 1, ascending = False)| 0 | 1 | |
|---|---|---|
| 30 | trimId | 3.681064e-01 |
| 21 | model_name | 2.059636e-01 |
| 23 | power | 1.224561e-01 |
| 20 | mileage | 8.733049e-02 |
| 29 | torque | 2.501834e-02 |
| 0 | back_legroom | 1.947877e-02 |
| 25 | savings_amount | 1.366988e-02 |
| 11 | height | 1.346977e-02 |
| 47 | engine_cylinders_V6 | 9.795723e-03 |
| 33 | year | 9.452366e-03 |
| 15 | latitude | 9.071714e-03 |
| 26 | seller_rating | 8.884898e-03 |
| 27 | sp_id | 8.311864e-03 |
| 16 | length | 8.101590e-03 |
| 12 | highway_fuel_economy | 8.033217e-03 |
| 32 | width | 7.344786e-03 |
| 9 | fuel_tank_volume | 7.089915e-03 |
| 2 | daysonmarket | 6.167395e-03 |
| 36 | wheel_system_cat | 6.084273e-03 |
| 18 | longitude | 5.956792e-03 |
| 37 | wheel_system_display_cat | 5.825781e-03 |
| 31 | wheelbase | 5.460589e-03 |
| 3 | dealer_zip | 5.311008e-03 |
| 8 | front_legroom | 5.194709e-03 |
| 17 | listed_date | 4.584625e-03 |
| 4 | engine_displacement | 4.355540e-03 |
| 1 | city_fuel_economy | 3.154374e-03 |
| 7 | franchise_make | 2.225961e-03 |
| 42 | body_type_SUV_Crossover | 1.811286e-03 |
| 22 | owner_count | 1.508617e-03 |
| 19 | maximum_seating | 1.295084e-03 |
| 51 | transmission_display_6_Speed_Automatic | 9.956286e-04 |
| 10 | has_accidents | 9.003557e-04 |
| 54 | transmission_display_Automatic | 8.624836e-04 |
| 13 | isCab | 8.238435e-04 |
| 5 | fleet | 7.241308e-04 |
| 52 | transmission_display_8_Speed_Automatic | 6.188337e-04 |
| 43 | body_type_Sedan | 5.821297e-04 |
| 46 | engine_cylinders_I4 | 5.186393e-04 |
| 34 | fuel_type_cat | 5.060303e-04 |
| 35 | transmission_cat | 5.033826e-04 |
| 55 | transmission_display_Continuously_Variable_Tra... | 4.730021e-04 |
| 56 | transmission_display_other | 3.798145e-04 |
| 45 | body_type_Wagon | 2.671794e-04 |
| 53 | transmission_display_9_Speed_Automatic | 2.053301e-04 |
| 48 | engine_cylinders_V6_Flex_Fuel_Vehicle | 1.959885e-04 |
| 39 | body_type_Hatchback | 1.790901e-04 |
| 14 | is_new | 1.749637e-04 |
| 41 | body_type_Pickup_Truck | 1.455960e-04 |
| 40 | body_type_Minivan | 1.367203e-04 |
| 50 | engine_cylinders_other | 1.187689e-04 |
| 49 | engine_cylinders_V8 | 8.934867e-05 |
| 38 | body_type_Coupe | 6.330553e-05 |
| 28 | theft_title | 1.080973e-05 |
| 6 | frame_damaged | 6.621246e-06 |
| 44 | body_type_Van | 2.167886e-06 |
| 24 | salvage | 4.001456e-07 |
rf_feats[(rf_feats[1] >= 0.2) | (rf_feats[1] <= -.2)]| 0 | 1 | |
|---|---|---|
| 21 | model_name | 0.205964 |
| 30 | trimId | 0.368106 |
base_dt = DecisionTreeRegressor(random_state = 4).fit(X_tree, y_tree)
dt_feats = pd.DataFrame(zip(base_dt.feature_names_in_, base_dt.feature_importances_))
dt_feats.sort_values(by = 1, ascending = False)
dt_feats[(dt_feats[1] >= 0.2) | (dt_feats[1] <= -.2)]| 0 | 1 | |
|---|---|---|
| 21 | model_name | 0.211757 |
| 30 | trimId | 0.379266 |
bagged_dt = BaggingRegressor(DecisionTreeRegressor(), random_state = 4, n_estimators = 100).fit(X_tree, y_tree)
f_imp = pd.DataFrame(zip(bagged_dt.feature_names_in_,np.mean([tree.feature_importances_ for tree in bagged_dt.estimators_], axis = 0)))f_imp.sort_values(by = 1, ascending = False)
f_imp[(f_imp[1] >= 0.2) | (f_imp[1] <= -.2)]| 0 | 1 | |
|---|---|---|
| 21 | model_name | 0.205927 |
| 30 | trimId | 0.368145 |
Based on the base Random Forest and DecisionTree, the two most significant predictors are model_name and trimId. We can test the performance of a model developed using our feature selection and a model developed using the selected features by the RandomForest to find the best predictors.
X_tree_subset = X_tree[['model_name', 'trimId']]
X_train_sub, X_test_sub, y_train_sub, y_test_sub = train_test_split(X_tree_subset, y_tree, test_size = 0.3, random_state = 4)Base MARS model on the tree selected features:
base_mars1 = Earth().fit(X_train_sub, y_train_sub)
print(base_mars1.summary())
print("Base MARS train RMSE", np.sqrt(base_mars1.mse_))
print("\nBase MARS train R^2", base_mars1.rsq_)
print("\nBASE MARS Test RMSE", mean_squared_error(y_test_sub, base_mars1.predict(X_test_sub), squared = False))
print("\nBASE MARS Test R^2", r2_score(y_test_sub, base_mars1.predict(X_test_sub)))Earth Model
--------------------------------------
Basis Function Pruned Coefficient
--------------------------------------
(Intercept) No 6604.08
h(trimId-67312) No -0.228409
h(67312-trimId) Yes None
model_name No 3970.31
h(trimId-32660) No 0.196858
h(32660-trimId) Yes None
h(trimId-83763) Yes None
h(83763-trimId) Yes None
h(trimId-89881) No -2.24943
h(89881-trimId) Yes None
h(trimId-78253) No -6.47566
h(78253-trimId) Yes None
h(trimId-76583) No 2.17212
h(76583-trimId) Yes None
h(trimId-78985) No 5.02076
h(78985-trimId) No -1.16045
h(trimId-82290) Yes None
h(82290-trimId) No 1.05506
trimId Yes None
--------------------------------------
MSE: 12520924.7798, GCV: 12709856.5249, RSQ: 0.6517, GRSQ: 0.6466
Base MARS train RMSE 3538.4918792925073
Base MARS train R^2 0.651676960724981
BASE MARS Test RMSE 3596.9618257602892
BASE MARS Test R^2 0.6314408357383765
Base MARS model on our selected features:
base_mars2 = Earth().fit(X_train, y_train)
print(base_mars2.summary())
print("Base MARS train RMSE", np.sqrt(base_mars2.mse_))
print("\nBase MARS train R^2", base_mars2.rsq_)
print("\nBASE MARS Test RMSE", mean_squared_error(y_test, base_mars2.predict(X_test), squared = False))
print("\nBASE MARS Test R^2", r2_score(y_test, base_mars2.predict(X_test)))Earth Model
-----------------------------------------------
Basis Function Pruned Coefficient
-----------------------------------------------
(Intercept) No -696322
h(trimId-67312) Yes None
h(67312-trimId) No -0.126974
model_name No 2187.29
power No 14.7161
h(mileage-116656) No -0.032541
h(116656-mileage) No 0.0639984
wheel_system_display_cat No -355.43
engine_cylinders_V6 No 1588.48
torque No 11.9225
h(trimId-37340) Yes None
h(37340-trimId) No 0.187547
wheel_system_cat No -399.524
h(savings_amount-4356) No 2.82244
h(4356-savings_amount) No -0.259224
year No 351.544
body_type_Sedan No -761.063
h(city_fuel_economy-32) Yes None
h(32-city_fuel_economy) No -334.968
length No 67.983
franchise_make No 223.224
highway_fuel_economy No -175.368
wheelbase No -67.8022
-----------------------------------------------
MSE: 7209072.4230, GCV: 7434760.4238, RSQ: 0.7982, GRSQ: 0.7921
Base MARS train RMSE 2684.9715869996703
Base MARS train R^2 0.7982451686817187
BASE MARS Test RMSE 2874.664784272254
BASE MARS Test R^2 0.7688754274179378
The model’s performance is much better using the features selected by our analysis than the tree based models. The R^2 is much greater on our subset’s test data even though there does appear to be some slight overfitting. Overfitting does not appear to be an issue with the tree based models selected features but the model’s performance is much worse.
The main takeaway from this analysis is that our selected features resulted in overfitting but generally well performing model. This was true for our initial model development in which most of our models overfit even with extensive tuning and generally had much higher RMSE’s at the base model and tuned model.
Another method for finding the best features is by running a base MARS model on our original data to find the best overall features.
X_tree.shape(4501, 57)
mars_feats = Earth().fit(X_tree, y_tree)
print(mars_feats.summary())Earth Model
-------------------------------------------------------------
Basis Function Pruned Coefficient
-------------------------------------------------------------
(Intercept) No -899382
h(trimId-67983) Yes None
h(67983-trimId) No -0.068928
model_name No 2020.83
power No 16.0908
h(mileage-117301) No -0.0349422
h(117301-mileage) No 0.0597747
wheel_system_cat No -358.023
engine_cylinders_V6_Flex_Fuel_Vehicle No -3210.91
fleet No -1076.01
h(savings_amount-1580) No 0.804517
h(1580-savings_amount) Yes None
h(latitude-30.4119) No -41.4752
h(30.4119-latitude) No -369.663
engine_cylinders_I4 No -865.66
wheel_system_display_cat No -246.752
h(trimId-35944) No 0.0450436
h(35944-trimId) No 0.114317
year No 455.331
transmission_display_6_Speed_Automatic No -724.604
h(daysonmarket-215) Yes None
h(215-daysonmarket) No 59.4532
highway_fuel_economy No -275.857
h(city_fuel_economy-32) No -783.609
h(32-city_fuel_economy) No -264.844
h(length-231.9) No -910.335
h(231.9-length) No -55.2657
seller_rating No 437.54
franchise_make No 192.154
h(daysonmarket-256) Yes None
h(256-daysonmarket) No -50.5369
torque No 9.83559
h(wheelbase-145) No 709.676
h(145-wheelbase) No 48.9592
-------------------------------------------------------------
MSE: 6766485.4663, GCV: 6993008.2321, RSQ: 0.8104, GRSQ: 0.8042
print("Base MARS train RMSE", np.sqrt(mars_feats.mse_))
print("\nBase MARS train R^2", mars_feats.rsq_)Base MARS train RMSE 2601.2469060613644
Base MARS train R^2 0.8104297884139036
print("Checking performance on split data: \n")
X_train_mars, X_test_mars, y_train_mars, y_test_mars = train_test_split(X_tree, y_tree, random_state = 4, test_size = 0.3)
random.seed(4)
mars_feats2 = Earth(feature_importance_type = 'rss').fit(X_train_mars, y_train_mars)
print("Base MARS Train RMSE:", np.sqrt(mars_feats2.mse_))
print("\nBase MARS Train R^2", mars_feats2.rsq_)
print("\nBase MARS Test RMSE:", mean_squared_error(y_test_mars, mars_feats2.predict(X_test_mars), squared = False))
print("\nBase MARS Test R^2", r2_score(y_test_mars, mars_feats2.predict(X_test_mars)))Checking performance on split data:
Base MARS Train RMSE: 2629.4932798721816
Base MARS Train R^2 0.8076510034139952
Base MARS Test RMSE: 2619.6932550051956
Base MARS Test R^2 0.8045046459515068
print(mars_feats2.summary_feature_importances(sort_by = 'rss')) rss
model_name 0.43
mileage 0.36
power 0.09
year 0.06
wheel_system_cat 0.02
engine_cylinders_V6_Flex_Fuel_Vehicle 0.01
fleet 0.01
savings_amount 0.01
engine_cylinders_I4 0.00
city_fuel_economy 0.00
trimId 0.00
listed_date 0.00
latitude 0.00
transmission_display_6_Speed_Automatic 0.00
seller_rating 0.00
highway_fuel_economy 0.00
franchise_make 0.00
length 0.00
wheelbase 0.00
wheel_system_display_cat 0.00
longitude 0.00
maximum_seating 0.00
transmission_display_other 0.00
is_new 0.00
owner_count 0.00
height 0.00
has_accidents 0.00
fuel_tank_volume 0.00
front_legroom 0.00
frame_damaged 0.00
engine_displacement 0.00
dealer_zip 0.00
daysonmarket 0.00
isCab 0.00
theft_title 0.00
salvage 0.00
sp_id 0.00
transmission_display_Automatic 0.00
transmission_display_9_Speed_Automatic 0.00
transmission_display_8_Speed_Automatic 0.00
engine_cylinders_other 0.00
engine_cylinders_V8 0.00
engine_cylinders_V6 0.00
body_type_Wagon 0.00
body_type_Van 0.00
body_type_Sedan 0.00
body_type_SUV_Crossover 0.00
body_type_Pickup_Truck 0.00
body_type_Minivan 0.00
body_type_Hatchback 0.00
body_type_Coupe 0.00
transmission_cat 0.00
fuel_type_cat 0.00
width 0.00
torque 0.00
transmission_display_Continuously_Variable_Transmission 0.00
back_legroom 0.00
The selected features by MARS result in the best performance on the training data and the testing data. There is no overfitting or underfitting and the R^2 has improved, albeit marginally.
Thus, we will perform the rest of our analysis with the top features selected by the best MARS model. These features are:
model_name : 0.43
mileage 0.36
power 0.09
year 0.06
wheel_system_cat 0.02
engine_cylinders_V6_Flex_Fuel_Vehicle 0.01
fleet 0.01
savings_amount 0.01
Final data subsetting and split:
# dropping the 31 missing values from the main/original data
data.dropna(inplace = True)
y_mars = data.price
data.drop(columns = ['price'], inplace = True)
print("Final overall training data shape", data.shape)
print("Final response variable shape", y_mars.shape)
# subsetting for the final selected features
X_mars= data[['model_name', 'mileage', 'power', 'year', 'wheel_system_cat', 'engine_cylinders_V6_Flex_Fuel_Vehicle', 'fleet', 'savings_amount']]
# splitting the final data!
X_train_mars, X_test_mars, y_train_mars, y_test_mars = train_test_split(X_mars, y_mars, test_size = 0.3, random_state = 4)Final overall training data shape (4501, 57)
Final response variable shape (4501,)
We will develop our models using both the features found during our EDA and the models using the features found based on the MARS model analysis.
2 Developing the model: Hyperparameter tuning
Throughout our model development, we ran our analysis on both our selected features and the random forest’s selected features to observe the difference in performance. Generally, we anticipate that the models developed using the features selected by random forest will perform better.
X_train, y_train, X_test, and y_test represent the data which contains the feature selection resulting from our EDA and analysis while X_train_mars, y_train_mars, X_test_mars, and y_test_mars represent the data which contains the features selected by the Random Forest model.
2.1 MARS & Bagging MARS
By Hiba Khatib
2.1.1 Tuning MARS models
Since we already performed base MARS model analysis to inform our feature selection, we can continue with tuning MARS to achieve better results. The first set of tuning will be on the MARS selected features then we will also tuning on our selected features.
random.seed(4)
opt_mars = pd.DataFrame(columns = {'degree', 'R2', 'RMSE'})
iter_num = 0
cv = KFold(n_splits=5, random_state=4, shuffle=True)
for i in range(1, 8):
MARS_model = Earth(max_terms=500, max_degree=i)
r2 = cross_val_score(MARS_model, X_train_mars, y_train_mars, cv=5, scoring='r2')
mean_r2 = r2.mean()
rmse = cross_val_score(MARS_model, X_train_mars, y_train_mars, cv= 5, scoring='neg_root_mean_squared_error')
mean_rmse = abs(rmse).mean()
opt_mars.loc[iter_num, 'degree'] = i
opt_mars.loc[iter_num, 'R2'] = mean_r2
opt_mars.loc[iter_num, 'RMSE'] = mean_rmse
iter_num += 1opt_mars.sort_values(by = 'R2', ascending = False).head(1)| R2 | RMSE | degree | |
|---|---|---|---|
| 4 | 0.790333 | 2742.648575 | 5 |
opt_mars.sort_values(by = 'RMSE', ascending = True).head(1)| R2 | RMSE | degree | |
|---|---|---|---|
| 4 | 0.790333 | 2742.648575 | 5 |
The optimal degree is 5 for both R2 and RMSE.
fit_mars1 = Earth(max_terms=1000, max_degree= 5, feature_importance_type = "rss").fit(X_train_mars, y_train_mars)
print(fit_mars1.summary())Earth Model
----------------------------------------------------------------------------------------------
Basis Function Pruned Coefficient
----------------------------------------------------------------------------------------------
(Intercept) No -0.867045
year No -872.929
model_name*year No 1.1003
power*year No 0.0247296
h(mileage-118799)*year No 3.49146e-05
h(118799-mileage)*year Yes None
wheel_system_cat*year No -0.432392
engine_cylinders_V6_Flex_Fuel_Vehicle*model_name*year Yes None
h(mileage-86269)*model_name*year No -8.66159e-06
h(86269-mileage)*model_name*year Yes None
fleet*power*year No -0.00229866
year*year No 0.435352
h(savings_amount-1293) No 0.737447
h(1293-savings_amount) Yes None
power*h(86269-mileage)*model_name*year No 4.76397e-06
mileage*power*year No -1.1364e-07
wheel_system_cat*engine_cylinders_V6_Flex_Fuel_Vehicle*model_name*year No -0.503445
power*power*h(86269-mileage)*model_name*year No -4.14987e-10
year*power*h(86269-mileage)*model_name*year No -2.30134e-09
----------------------------------------------------------------------------------------------
MSE: 7320763.8309, GCV: 7491008.4641, RSQ: 0.7963, GRSQ: 0.7917
print("Performance on MARS feature selection:\n")
print("Tuned MARS Train RMSE:", np.sqrt(fit_mars1.mse_))
print("Tuned MARS Train R^2 value: ", fit_mars1.rsq_)
print("Tuned MARS Test RMSE:",mean_squared_error(y_test_mars, fit_mars1.predict(X_test_mars), squared = False) )
print("Tuned MARS Test R^2:", r2_score(y_test_mars, fit_mars1.predict(X_test_mars)))Performance on MARS feature selection:
Tuned MARS Train RMSE: 2705.691008032262
Tuned MARS Train R^2 value: 0.7963416638745415
Tuned MARS Test RMSE: 2679.9773361980965
Tuned MARS Test R^2: 0.7954036885971033
print("Feature importance by RSS on MARS feature selection\n")
print(fit_mars1.summary_feature_importances(sort_by = 'rss'))Feature importance by RSS on MARS feature selection
rss
year 0.59
model_name 0.24
power 0.09
mileage 0.05
wheel_system_cat 0.02
engine_cylinders_V6_Flex_Fuel_Vehicle 0.01
fleet 0.00
savings_amount 0.00
Performance on our feature selection:
random.seed(4)
opt_mars2 = pd.DataFrame(columns = {'degree', 'R2', 'RMSE'})
iter_num = 0
cv = KFold(n_splits=5, random_state=4, shuffle=True)
for i in range(1, 8):
MARS_model = Earth(max_terms=500, max_degree=i)
r2 = cross_val_score(MARS_model, X_train, y_train, cv=5, scoring='r2')
mean_r2 = r2.mean()
rmse = cross_val_score(MARS_model, X_train, y_train, cv= 5, scoring='neg_root_mean_squared_error')
mean_rmse = abs(rmse).mean()
opt_mars2.loc[iter_num, 'degree'] = i
opt_mars2.loc[iter_num, 'R2'] = mean_r2
opt_mars2.loc[iter_num, 'RMSE'] = mean_rmse
iter_num += 1print(opt_mars2.sort_values(by = 'R2', ascending = False).head(1))
print(opt_mars2.sort_values(by = 'RMSE', ascending = True).head(1)) R2 RMSE degree
5 0.81581 2560.543379 6
R2 RMSE degree
5 0.81581 2560.543379 6
The best degree for both R^2 and RMSE is 6
fit_mars2 = Earth(max_terms=1000, max_degree= 6, feature_importance_type = "rss").fit(X_train, y_train)
print(fit_mars2.summary())Earth Model
----------------------------------------------------------------------------
Basis Function Pruned Coefficient
----------------------------------------------------------------------------
(Intercept) No 7265.17
h(trimId-67312) No -0.385114
h(67312-trimId) No -0.0784129
model_name No -460005
power No 50.5903
h(mileage-116656) No -0.0256394
h(116656-mileage) No 0.0538941
wheel_system_display_cat Yes None
h(trimId-60496)*h(67312-trimId) No 0.00181749
h(60496-trimId)*h(67312-trimId) Yes None
h(trimId-67312)*power No 0.00396033
h(67312-trimId)*power No -0.000471043
power*h(trimId-67312)*power No -1.83312e-05
city_fuel_economy*power*h(trimId-67312)*power No 4.80116e-07
highway_fuel_economy*power No -0.7356
engine_cylinders_V6*h(116656-mileage) Yes None
torque No 13.6581
h(trimId-16075)*wheel_system_display_cat No -0.0120157
h(16075-trimId)*wheel_system_display_cat Yes None
h(city_fuel_economy-32)*highway_fuel_economy*power Yes None
h(32-city_fuel_economy)*highway_fuel_economy*power No -0.0412428
h(savings_amount-1465)*wheel_system_display_cat No 0.262227
h(1465-savings_amount)*wheel_system_display_cat Yes None
year*model_name No 229.382
model_name*engine_cylinders_V6*h(116656-mileage) No -0.0168493
engine_displacement*h(trimId-60496)*h(67312-trimId) No 3.33867e-07
model_name*h(16075-trimId)*wheel_system_display_cat No 0.161783
height*h(trimId-60496)*h(67312-trimId) No -4.5658e-05
height*engine_cylinders_V6*h(116656-mileage) No 0.000785742
franchise_make*h(116656-mileage) No 0.00241208
h(engine_displacement-6000)*wheel_system_display_cat No 6.56446
h(6000-engine_displacement)*wheel_system_display_cat Yes None
----------------------------------------------------------------------------
MSE: 6085273.5118, GCV: 6326329.5161, RSQ: 0.8297, GRSQ: 0.8231
print("Performance on MARS feature selection:\n")
print("Tuned MARS Train RMSE:", np.sqrt(fit_mars2.mse_))
print("Tuned MARS Train R^2 value: ", fit_mars2.rsq_)
print("Tuned MARS Test RMSE:",mean_squared_error(y_test, fit_mars2.predict(X_test), squared = False) )
print("Tuned MARS Test R^2:", r2_score(y_test, fit_mars2.predict(X_test)))Performance on MARS feature selection:
Tuned MARS Train RMSE: 2466.834715144428
Tuned MARS Train R^2 value: 0.829696074769481
Tuned MARS Test RMSE: 2743.978978341708
Tuned MARS Test R^2: 0.7894121729348178
print("Feature importance by RSS on MARS feature selection\n")
print(fit_mars2.summary_feature_importances(sort_by = 'rss'))Feature importance by RSS on MARS feature selection
rss
model_name 0.27
year 0.26
mileage 0.24
power 0.11
trimId 0.07
wheel_system_display_cat 0.02
engine_cylinders_V6 0.01
highway_fuel_economy 0.01
height 0.01
city_fuel_economy 0.00
engine_displacement 0.00
savings_amount 0.00
torque 0.00
franchise_make 0.00
owner_count 0.00
wheelbase 0.00
width 0.00
length 0.00
wheel_system_cat 0.00
body_type_Sedan 0.00
fuel_tank_volume 0.00
engine_cylinders_I4 0.00
transmission_display_8_Speed_Automatic 0.00
Results and Takeaways:
The MARS selected features model seems to outperform the model developed on our subset. There is less overfitting despite the slightly worse R^2. This makes sense because the original MARS feature selection ensured that we kept the features which best represented the data and thus it is less likely to experience over or underfitting as in our selected features. The pruning ensures that the model is performing as optimally as possible with the base parameters.
It is also important to note that the performance on the model with the tuned degree is not significantly better than the performance on the base MARS model with the MARS selected features.
2.1.2 Bagging MARS
First we will bag the best features selected by MARS. We will do a base MARS fitted with the MARS feature data and then a the best MARS feature.
# bagging the base model for reference
bagging_base = BaggingRegressor(base_estimator = Earth(feature_importance_type = 'rss'),
random_state = 4).fit(X_train_mars, y_train_mars)
best_rmse = 10000
best_r2 = 0
for estimator in bagging_base.estimators_:
if np.sqrt(estimator.mse_) < best_rmse:
best_rmse = np.sqrt(estimator.mse_)
if estimator.rsq_ > best_r2:
best_r2 = estimator.rsq_
print("Bagged base MARS model Train RMSE:", best_rmse)
print("R^2 on Train data", best_r2)
print("Bagged base MARS model Test RMSE:", mean_squared_error(y_test_mars, bagging_base.predict(X_test_mars), squared = False))
print("R^2 on Test data:", r2_score(y_test_mars, bagging_base.predict(X_test_mars)))
print("Feature importances show the same feature importances as the unbagged MARS models.")
# for estimator in bagging_base.estimators_:
# print(estimator.summary_feature_importances(sort_by = "rss"))Bagged base MARS model Train RMSE: 2761.6200070905847
R^2 on Train data 0.9588173903334842
Bagged base MARS model Test RMSE: 2788.600146839992
R^2 on Test data: 0.7784824962191642
Feature importances show the same feature importances as the unbagged MARS models.
Now we will bag the best MARS model found on the MARS selected features and also tune the bagging grid for this model.
# bagging the best estimator found on the MARS selected features
bagging_mars = BaggingRegressor(base_estimator = Earth(max_terms=1000, max_degree= 5, feature_importance_type = "rss")).fit(X_train_mars, y_train_mars)
best_rmse = 10000
best_r2 = 0
for estimator in bagging_mars.estimators_:
if np.sqrt(estimator.mse_) < best_rmse:
best_rmse = np.sqrt(estimator.mse_)
if estimator.rsq_ > best_r2:
best_r2 = estimator.rsq_
print("Bagged base MARS model Train RMSE:", best_rmse)
print("R^2 on Train data", best_r2)
print("Bagged base MARS model Test RMSE:", mean_squared_error(y_test_mars, bagging_mars.predict(X_test_mars), squared = False))
print("R^2 on Test data:", r2_score(y_test_mars, bagging_mars.predict(X_test_mars)))
print("Feature importances are consistent.")Bagged base MARS model Train RMSE: 2612.6987983602867
R^2 on Train data 0.963612437865765
Bagged base MARS model Test RMSE: 2737.9654132203773
R^2 on Test data: 0.7864539845171037
Feature importances are consistent.
# coarse grid search: optimizing bagging model
bagging_opt = {'bootstrap': [True, False],
'bootstrap_features': [True, False],
'max_samples': [0.5, 0.75,1.0],
'max_features': [0.5, 0.75,1.0]}
opt_grid = RandomizedSearchCV(estimator = BaggingRegressor(base_estimator = Earth(max_degree= 5, feature_importance_type = "rss"), random_state =4), \
param_distributions = bagging_opt, verbose = 1, n_jobs = -1, scoring = 'neg_root_mean_squared_error')
opt_grid.fit(X_train_mars, y_train_mars)Fitting 5 folds for each of 10 candidates, totalling 50 fits
RandomizedSearchCV(estimator=BaggingRegressor(base_estimator=Earth(feature_importance_type='rss',
max_degree=5),
random_state=4),
n_jobs=-1,
param_distributions={'bootstrap': [True, False],
'bootstrap_features': [True, False],
'max_features': [0.5, 0.75, 1.0],
'max_samples': [0.5, 0.75, 1.0]},
scoring='neg_root_mean_squared_error', verbose=1)
print("The best BaggingRegressor coarse grid parameters:", opt_grid.best_params_)
best_rmse = 10000
best_r2 = 0
for estimator in opt_grid.best_estimator_.estimators_:
if np.sqrt(estimator.mse_) < best_rmse:
best_rmse = np.sqrt(estimator.mse_)
if estimator.rsq_ > best_r2:
best_r2 = estimator.rsq_
print("Tuned BaggingRegressor Train RMSE:", best_rmse)
print("Tuned BaggingRegressor R2:", best_r2)
print("Tuned BaggingRegressor Test RMSE: ", mean_squared_error(y_test_mars, opt_grid.predict(X_test_mars), squared = False))
print("Tuned BaggingRegressor R2", r2_score(y_test_mars, opt_grid.predict(X_test_mars)))The best BaggingRegressor coarse grid parameters: {'max_samples': 0.75, 'max_features': 0.5, 'bootstrap_features': False, 'bootstrap': False}
Tuned BaggingRegressor Train RMSE: 3040.6124116396736
Tuned BaggingRegressor R2: 0.9284106503760392
Tuned BaggingRegressor Test RMSE: 2914.9556727676986
Tuned BaggingRegressor R2 0.7579531323185931
# fine grid search
bagging_opt2 = {'max_samples': np.arange(0.1, 0.75, .2),
'max_features': np.arange(0.5, 1, .2)}
opt_grid2 = RandomizedSearchCV(estimator = BaggingRegressor(base_estimator = Earth(max_degree= 5, feature_importance_type = "rss"), \
random_state =4, bootstrap_features = False,\
bootstrap = False), param_distributions = bagging_opt2, verbose = 1,\
n_jobs = -1, scoring = 'neg_root_mean_squared_error', random_state = 4, n_iter = 3)
opt_grid2.fit(X_train_mars, y_train_mars)print("The best BaggingRegressor fine parameters:", opt_grid2.best_params_)
best_rmse = 10000
best_r2 = 0
for estimator in opt_grid2.best_estimator_.estimators_:
if np.sqrt(estimator.mse_) < best_rmse:
best_rmse = np.sqrt(estimator.mse_)
if estimator.rsq_ > best_r2:
best_r2 = estimator.rsq_
print("Tuned BaggingRegressor Train RMSE:", best_rmse)
print("Tuned BaggingRegressor R2:", best_r2)
print("Tuned BaggingRegressor Test RMSE: ", mean_squared_error(y_test_mars, opt_grid2.predict(X_test_mars), squared = False))
print("Tuned BaggingRegressor R2", r2_score(y_test_mars, opt_grid2.predict(X_test_mars)))The best BaggingRegressor fine parameters: {'max_samples': 0.6, 'max_features': 0.9}
Tuned BaggingRegressor Train RMSE: 2620.6858359626726
Tuned BaggingRegressor R2: 0.9624349584512664
Tuned BaggingRegressor Test RMSE: 3145.216344284546
Tuned BaggingRegressor R2 0.718202844680064
Finally we will bag the best MARS model found on our selected features and also tune the bagging grid for this model.
# bagging the base model for reference
bagging_base = BaggingRegressor(base_estimator = Earth(feature_importance_type = 'rss'),
random_state = 4).fit(X_train, y_train)
best_rmse = 10000
best_r2 = 0
for estimator in bagging_base.estimators_:
if np.sqrt(estimator.mse_) < best_rmse:
best_rmse = np.sqrt(estimator.mse_)
if estimator.rsq_ > best_r2:
best_r2 = estimator.rsq_
print("Bagged base MARS model Train RMSE:", best_rmse)
print("R^2 on Train data", best_r2)
print("Bagged base MARS model Test RMSE:", mean_squared_error(y_test, bagging_base.predict(X_test), squared = False))
print("R^2 on Test data:", r2_score(y_test, bagging_base.predict(X_test)))
print("Feature importances show the same feature importances as the unbagged MARS models.")
# for estimator in bagging_base.estimators_:
# print(estimator.summary_feature_importances(sort_by = "rss"))Bagged base MARS model Train RMSE: 2597.9794766802843
R^2 on Train data 0.9632264134685602
Bagged base MARS model Test RMSE: 2869.7450024161185
R^2 on Test data: 0.7696658565959801
Feature importances show the same feature importances as the unbagged MARS models.
# bagging the best estimator found on our selected features
bagging_mars = BaggingRegressor(base_estimator = Earth(max_degree= 6, feature_importance_type = "rss")).fit(X_train, y_train)
best_rmse = 10000
best_r2 = 0
for estimator in bagging_mars.estimators_:
if np.sqrt(estimator.mse_) < best_rmse:
best_rmse = np.sqrt(estimator.mse_)
if estimator.rsq_ > best_r2:
best_r2 = estimator.rsq_
print("Bagged base MARS model Train RMSE:", best_rmse)
print("R^2 on Train data", best_r2)
print("Bagged base MARS model Test RMSE:", mean_squared_error(y_test, bagging_mars.predict(X_test), squared = False))
print("R^2 on Test data:", r2_score(y_test, bagging_mars.predict(X_test)))
print("Feature importances are consistent.")Bagged base MARS model Train RMSE: 2280.922799036446
R^2 on Train data 0.9720576756220233
Bagged base MARS model Test RMSE: 2940.2215805350525
R^2 on Test data: 0.7582136243310234
Feature importances are consistent.
# coarse grid search: optimizing bagging model
bagging_opt = {'max_samples': [0.5, 0.75,1.0],
'max_features': [0.5, 0.75,1.0]}
opt_grid = RandomizedSearchCV(estimator = BaggingRegressor(base_estimator = Earth(max_degree= 6, feature_importance_type = "rss"), random_state =4), \
param_distributions = bagging_opt, verbose = 1, n_jobs = -1, scoring = 'neg_root_mean_squared_error')
opt_grid.fit(X_train, y_train)Fitting 5 folds for each of 9 candidates, totalling 45 fits
RandomizedSearchCV(estimator=BaggingRegressor(base_estimator=Earth(feature_importance_type='rss',
max_degree=6),
random_state=4),
n_jobs=-1,
param_distributions={'max_features': [0.5, 0.75, 1.0],
'max_samples': [0.5, 0.75, 1.0]},
scoring='neg_root_mean_squared_error', verbose=1)
print("The best BaggingRegressor coarse grid parameters:", opt_grid.best_params_)
best_rmse = 10000
best_r2 = 0
for estimator in opt_grid.best_estimator_.estimators_:
if np.sqrt(estimator.mse_) < best_rmse:
best_rmse = np.sqrt(estimator.mse_)
if estimator.rsq_ > best_r2:
best_r2 = estimator.rsq_The best BaggingRegressor coarse grid parameters: {'max_samples': 1.0, 'max_features': 1.0}
print("Tuned BaggingRegressor Train RMSE:", best_rmse)
print("Tuned BaggingRegressor R2:", best_r2)
print("Tuned BaggingRegressor Test RMSE: ", mean_squared_error(y_test, opt_grid.predict(X_test), squared = False))
print("Tuned BaggingRegressor R2", r2_score(y_test, opt_grid.predict(X_test)))Tuned BaggingRegressor Train RMSE: 2263.553508108404
Tuned BaggingRegressor R2: 0.9721508909839748
Tuned BaggingRegressor Test RMSE: 2615.7244744165496
Tuned BaggingRegressor R2 0.8086380025887268
Analysis of Results
The best RMSE and R2 results from the tuned bagging model on the best MARS model. The best RMSE is 2615.7 and the best R2 is 0.81
2.2 Decision Trees & Bagging Decision Trees
By Nicole Birova
2.2.1 Base Decision (Regression) Tree
base = DecisionTreeRegressor(random_state = 4, max_depth = 3)
base.fit(X_train_mars, y_train_mars)
y_pred = base.predict(X_test_mars)
mse = mean_squared_error(y_test_mars, y_pred)
rmse = np.sqrt(mean_squared_error(y_test_mars, y_pred))
print("Mean Squared Error:", mse)
print("RMSE: ", rmse)Mean Squared Error: 15049713.713885518
RMSE: 3879.3960501456304
#Libraries for visualizing trees
from sklearn.tree import export_graphviz
from six import StringIO
from IPython.display import Image
import pydotplus
import time as tm
#Visualizing the regression tree
dot_data = StringIO()
export_graphviz(base, out_file=dot_data,
filled=True, rounded=True,
feature_names =['model_name', 'mileage', 'power', 'year', 'wheel_system_cat', 'engine_cylinders_V6_Flex_Fuel_Vehicle', 'fleet', 'savings_amount'])
graph = pydotplus.graph_from_dot_data(dot_data.getvalue())
graph.write_png('car_price_tree.png')
Image(graph.create_png())2.2.2 Coarse Grid Tuning
model = DecisionTreeRegressor(random_state=4)
model.fit(X, y)
print("Maximum tree depth =", model.get_depth())
print("Maximum leaves =", model.get_n_leaves())Maximum tree depth = 32
Maximum leaves = 4485
#Finding cross-validation error for trees
parameters = {'max_depth':range(2,35, 3),'max_leaf_nodes':range(2,5000, 100)}
cv = KFold(n_splits = 5,shuffle=True,random_state=4)
model = GridSearchCV(DecisionTreeRegressor(random_state=4), parameters, n_jobs=-1,verbose=1,cv=cv,
scoring=['neg_mean_squared_error', 'r2'], refit = 'neg_mean_squared_error')
model.fit(X, y)
print (model.best_score_, model.best_params_) Fitting 5 folds for each of 550 candidates, totalling 2750 fits
-7826521.035438128 {'max_depth': 17, 'max_leaf_nodes': 202}
From the above cross-validation, the optimal hyperparameter values are max_depth = 17 and max_leaf_nodes = 202.
#Detailed results of k-fold cross validation
cv_results = pd.DataFrame(model.cv_results_)
#cv_results.head()fig, axes = plt.subplots(1,2,figsize=(14,5))
plt.subplots_adjust(wspace=0.2)
axes[0].plot(cv_results.param_max_depth, np.sqrt(-cv_results.mean_test_neg_mean_squared_error), 'o')
axes[0].set_ylim([2750, 3250])
axes[0].set_xlabel('Depth')
axes[0].set_ylabel('K-fold RMSE')
axes[1].plot(cv_results.param_max_leaf_nodes, np.sqrt(-cv_results.mean_test_neg_mean_squared_error), 'o')
axes[1].set_ylim([2750, 3050])
axes[1].set_xlabel('Leaves')
axes[1].set_ylabel('K-fold RMSE');We observe that for a depth of around 10 to 15, and number of leaves within 1500, we get the lowest K-fold RMSE. So, we do a finer search in that region to obtain more precise hyperparameter values.
#Developing the tree based on optimal hyperparameters found by Course Grid
model = DecisionTreeRegressor(random_state=4, max_depth=17,max_leaf_nodes=202)
model.fit(X_train, y_train)DecisionTreeRegressor(max_depth=17, max_leaf_nodes=202, random_state=4)
#RMSE on test data for Course Grid
np.sqrt(mean_squared_error(y_test, model.predict(X_test)))2937.5060204053157
2.2.3 Fine Grid Search
#Finding cross-validation error for trees
#start_time = tm.time()
parameters = {'max_depth':range(10,15),'max_leaf_nodes':range(2,1500)}
cv = KFold(n_splits = 5,shuffle=True,random_state=4)
model = GridSearchCV(DecisionTreeRegressor(random_state=4), parameters, n_jobs=-1,verbose=1,cv=cv)
model.fit(X_train, y_train)
print (model.best_score_, model.best_params_)
#print("Time taken =", round((tm.time() - start_time)/60), "minutes")Fitting 5 folds for each of 7490 candidates, totalling 37450 fits
0.7724882142563047 {'max_depth': 11, 'max_leaf_nodes': 110}
From the above cross-validation, the optimal hyperparameter values are max_depth = 12 and max_leaf_nodes = 263.
#Developing the tree based on optimal hyperparameters found by cross-validation
model = DecisionTreeRegressor(random_state=4, max_depth=12,max_leaf_nodes=263)
model.fit(X_train, y_train)DecisionTreeRegressor(max_depth=12, max_leaf_nodes=263, random_state=4)
#RMSE on test data
np.sqrt(mean_squared_error(y_test, model.predict(X_test)))2962.0109038778355
#Print feature importances
importance = model.feature_importances_
sorted_importance = sorted(zip(X_train.columns, importance), key=lambda x: x[1], reverse=True)
for feature, importance_score in sorted_importance:
print(f"{feature}: {importance_score}")trimId: 0.4075585469194736
model_name: 0.2273241878271935
power: 0.12637431177551994
mileage: 0.09104928854570515
torque: 0.03393322434031242
fuel_tank_volume: 0.012618293929088805
year: 0.012253314042672001
savings_amount: 0.011843906148941586
wheel_system_display_cat: 0.010638324267882195
engine_cylinders_V6: 0.010090434253703628
height: 0.009959776640979606
width: 0.009866082385388653
length: 0.00968792453468887
wheelbase: 0.006217915176717917
wheel_system_cat: 0.005686509998066591
engine_displacement: 0.004713117562936892
highway_fuel_economy: 0.004337185767229553
franchise_make: 0.002359356431146806
city_fuel_economy: 0.0014987990012930014
transmission_display_8_Speed_Automatic: 0.0008723331973126079
owner_count: 0.0007321715045578808
body_type_Sedan: 0.0003849957491886656
engine_cylinders_I4: 0.0
Similar to the other models, timID and model_name are the most important predictors.
2.2.4 Cost-Complexity Pruning
CCPmodel = DecisionTreeRegressor(random_state = 4)#model without any restrictions
path= CCPmodel.cost_complexity_pruning_path(X_train,y_train)# Compute pruning pathalphas=path['ccp_alphas']
len(alphas)2886
#start_time = tm.time()
cv = KFold(n_splits = 5,shuffle=True,random_state=1)
tree = GridSearchCV(DecisionTreeRegressor(random_state=4), param_grid = {'ccp_alpha':alphas},
scoring = 'neg_mean_squared_error',n_jobs=-1,verbose=1,cv=cv)
tree.fit(X_train, y_train)
print (tree.best_score_, tree.best_params_)
#print("Time taken =",round((tm.time()-start_time)/60), "minutes")Fitting 5 folds for each of 2886 candidates, totalling 14430 fits
-7937206.2704411475 {'ccp_alpha': 31132.093956765573}
model = DecisionTreeRegressor(ccp_alpha=16584.831558986974,random_state=4)
model.fit(X_train, y_train)
pred = model.predict(X_test)
np.sqrt(mean_squared_error(y_test, pred))2924.7599538817894
gridcv_results = pd.DataFrame(tree.cv_results_)
cv_error = -gridcv_results['mean_test_score']
#Visualizing the 5-fold cross validation error vs alpha
plt.plot(alphas,cv_error)
plt.xscale('log')
plt.xlabel('alpha')
plt.ylabel('K-fold MSE');#Zooming in to see the alpha where the 5-fold cross validation error is minimizing
plt.plot(alphas[0:4000],cv_error[0:4000])
plt.xlabel('alpha')
plt.ylabel('K-fold MSE');#stime = tm.time()
trees=[]
for i in alphas:
tree = DecisionTreeRegressor(ccp_alpha=i,random_state=4)
tree.fit(X_train, y_train)
trees.append(tree)
#print(tm.time()-stime)node_counts = [clf.tree_.node_count for clf in trees]
depth = [clf.tree_.max_depth for clf in trees]
fig, ax = plt.subplots(1, 2,figsize=(10,6))
ax[0].plot(alphas[0:5000], node_counts[0:5000], marker="o", drawstyle="steps-post")
ax[0].set_xlabel("alpha")
ax[0].set_ylabel("number of nodes")
ax[0].set_title("Number of nodes vs alpha")
ax[1].plot(alphas[0:5500], depth[0:5500], marker="o", drawstyle="steps-post")
ax[1].set_xlabel("alpha")
ax[1].set_ylabel("depth of tree")
ax[1].set_title("Depth vs alpha")Text(0.5, 1.0, 'Depth vs alpha')
train_scores = [clf.score(X_train, y_train) for clf in trees]
test_scores = [clf.score(X_test, y_test) for clf in trees]fig, ax = plt.subplots()
ax.set_xlabel("alpha")
ax.set_ylabel("accuracy")
ax.set_title("Accuracy vs alpha for training and testing sets")
ax.plot(alphas[0:5450], train_scores[0:5450], marker="o", label="train", drawstyle="steps-post")#Plotting the zoomed-in plot (ignoring very high alphas), otherwise it is hard to see the trend
ax.plot(alphas[0:5450], test_scores[0:5450], marker="o", label="test", drawstyle="steps-post")#Plotting the zoomed-in plot (ignoring very high alphas), otherwise it is hard to see the trend
ax.legend()
plt.show()2.2.5 Bagging
#Bagging the results of 1 decision tree (just to see impact in RMSE)
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=1, random_state=4,n_jobs=-1).fit(X_train, y_train)np.sqrt(mean_squared_error(y_test, model.predict(X_test)))3508.2134690063613
#Bagging the results of 10 decision trees
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=10, random_state=4,n_jobs=-1).fit(X_train, y_train)np.sqrt(mean_squared_error(y_test, model.predict(X_test)))2427.7266689078633
#Bagging the results of 100 decision trees
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=100, random_state=4,n_jobs=-1).fit(X_train, y_train)np.sqrt(mean_squared_error(y_test, model.predict(X_test)))2316.262383026707
#Finding model accuracy vs number of trees
oob_rsquared={};test_rsquared={};oob_rmse={};test_rmse = {}
for i in np.linspace(10,400,40,dtype=int):
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=i, random_state=4,
n_jobs=-1,oob_score=True).fit(X, y)
oob_rsquared[i]=model.oob_score_ #Returns the out-of_bag R-squared
test_rsquared[i]=model.score(X_test,y_test) #Returns the test R-squared
oob_rmse[i]=np.sqrt(mean_squared_error(model.oob_prediction_,y))
test_rmse[i]=np.sqrt(mean_squared_error(model.predict(X_test),y_test))plt.rcParams.update({'font.size': 15})
plt.figure(figsize=(8, 6), dpi=80)
plt.plot(oob_rsquared.keys(),oob_rsquared.values(),label = 'Out of bag R-squared')
plt.plot(oob_rsquared.keys(),oob_rsquared.values(),'o',color = 'blue')
plt.plot(test_rsquared.keys(),test_rsquared.values(), label = 'Test data R-squared')
plt.xlabel('Number of trees')
plt.ylabel('Rsquared')
plt.legend()<matplotlib.legend.Legend at 0x7fb5a96595e0>
#Visualizing out-of-bag RMSE and test data RMSE
plt.rcParams.update({'font.size': 15})
plt.figure(figsize=(8, 6), dpi=80)
plt.plot(oob_rmse.keys(),oob_rmse.values(),label = 'Out of bag RMSE')
plt.plot(oob_rmse.keys(),oob_rmse.values(),'o',color = 'blue')
plt.plot(test_rmse.keys(),test_rmse.values(), label = 'Test data RMSE')
plt.xlabel('Number of trees')
plt.ylabel('RMSE')
plt.legend()<matplotlib.legend.Legend at 0x7fb5a971cc40>
The out-of-bag R-squared initially increases, and then stabilizes after a certain number of trees (around 50 in this case). Similarly, the RMSE follows the same trend.
#Bagging with 50 trees
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=50, random_state=4,
oob_score=True,n_jobs=-1).fit(X_train, y_train)model.oob_score_0.862093480612047
#RMSE on test data
pred = model.predict(X_test)
np.sqrt(mean_squared_error(y_test, pred))2344.7245506909007
#Optimizing bagging hyperparameters using grid search
n_samples = X_train.shape[0]
n_features = 20
params = {'base_estimator': [DecisionTreeRegressor(random_state = 4),LinearRegression()],
'n_estimators': [50],
'max_samples': [0.5,1.0],
'max_features': [0.5,1.0],
'bootstrap': [True, False],
'bootstrap_features': [True, False]}
cv = KFold(n_splits=5,shuffle=True,random_state=1)
bagging_regressor_grid = GridSearchCV(BaggingRegressor(random_state=4, n_jobs=-1),
param_grid =params, cv=cv, n_jobs=-1, verbose=1)
bagging_regressor_grid.fit(X_train, y_train)
print('Train R^2 Score : %.3f'%bagging_regressor_grid.best_estimator_.score(X_train, y_train))
print('Test R^2 Score : %.3f'%bagging_regressor_grid.best_estimator_.score(X_test, y_test))
print('Best R^2 Score Through Grid Search : %.3f'%bagging_regressor_grid.best_score_)
print('Best Parameters : ',bagging_regressor_grid.best_params_)Fitting 5 folds for each of 32 candidates, totalling 160 fits
Train R^2 Score : 0.999
Test R^2 Score : 0.862
Best R^2 Score Through Grid Search : 0.867
Best Parameters : {'base_estimator': DecisionTreeRegressor(random_state=4), 'bootstrap': False, 'bootstrap_features': False, 'max_features': 0.5, 'max_samples': 1.0, 'n_estimators': 50}
#Model with optimal hyperparameters
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=50, random_state=4,
oob_score=True,n_jobs=-1,bootstrap_features=True,bootstrap=True,
max_features=1.0,max_samples=1.0).fit(X_train, y_train)#RMSE on test data
np.sqrt(mean_squared_error(y_test, model.predict(X_test)))2238.2224784779655
Bagging showed to be the best way to reduce RMSE, producing a lower RMSE than all the other decision tree models.
2.3 Random forest
By Hiba Khatib
The first set of Random Forests will be performed on the MARS selected features.
base_rf = RandomForestRegressor(random_state = 4).fit(X_train_mars, y_train_mars)
print("Base RandomForest Test RMSE: ", mean_squared_error(y_test_mars, base_rf.predict(X_test_mars), squared = False))
print("Base RandomForest Train RMSE: ", mean_squared_error(y_train_mars, base_rf.predict(X_train_mars), squared = False))
print("Base RandomForest R^2: ", r2_score(y_test_mars, base_rf.predict(X_test_mars)))Base RandomForest Test RMSE: 2369.927998736765
Base RandomForest Train RMSE: 936.6562391425674
Base RandomForest R^2: 0.840005202649853
base_rf_feats = pd.DataFrame(zip(base_rf.feature_names_in_, base_rf.feature_importances_)).reset_index()
base_rf_feats.sort_values(by = base_rf_feats.columns[2], ascending = False)| index | 0 | 1 | |
|---|---|---|---|
| 3 | 3 | year | 0.343984 |
| 0 | 0 | model_name | 0.242288 |
| 2 | 2 | power | 0.169672 |
| 1 | 1 | mileage | 0.147431 |
| 7 | 7 | savings_amount | 0.055077 |
| 4 | 4 | wheel_system_cat | 0.030605 |
| 6 | 6 | fleet | 0.005677 |
| 5 | 5 | engine_cylinders_V6_Flex_Fuel_Vehicle | 0.005266 |
2.3.1 Coarse grid search
rf_params = {'n_estimators': [250, 500, 700],
'max_depth': [4, 6, 8],
'max_leaf_nodes': range(2, 300, 75),
'bootstrap': [True, False]}
rf_grid = GridSearchCV(RandomForestRegressor(random_state = 4), rf_params, n_jobs = -1, verbose = 1,
scoring = 'neg_root_mean_squared_error')
rf_grid.fit(X_train_mars, y_train_mars)Fitting 5 folds for each of 72 candidates, totalling 360 fits
GridSearchCV(estimator=RandomForestRegressor(random_state=4), n_jobs=-1,
param_grid={'bootstrap': [True, False], 'max_depth': [4, 6, 8],
'max_leaf_nodes': range(2, 300, 75),
'n_estimators': [250, 500, 700]},
scoring='neg_root_mean_squared_error', verbose=1)
print("Best RandomForest parameters:", rf_grid.best_params_)
print("Best model train RMSE:", mean_squared_error(y_train_mars, rf_grid.predict(X_train_mars), squared = False))
print("Best model test RMSE:", mean_squared_error(y_test_mars, rf_grid.predict(X_test_mars), squared = False))
print("Best model R^2:", r2_score(y_test_mars, rf_grid.predict(X_test_mars)))Best RandomForest parameters: {'bootstrap': True, 'max_depth': 8, 'max_leaf_nodes': 152, 'n_estimators': 250}
Best model train RMSE: 2115.1675658763616
Best model test RMSE: 2512.4214163945476
Best model R^2: 0.8201872303352222
rf_feats = pd.DataFrame(zip(rf_grid.best_estimator_.feature_names_in_, rf_grid.best_estimator_.feature_importances_)).reset_index()
rf_feats.sort_values(by = rf_feats.columns[2], ascending = False)| index | 0 | 1 | |
|---|---|---|---|
| 3 | 3 | year | 0.381062 |
| 0 | 0 | model_name | 0.277125 |
| 2 | 2 | power | 0.157509 |
| 1 | 1 | mileage | 0.128131 |
| 4 | 4 | wheel_system_cat | 0.028307 |
| 7 | 7 | savings_amount | 0.020797 |
| 5 | 5 | engine_cylinders_V6_Flex_Fuel_Vehicle | 0.004691 |
| 6 | 6 | fleet | 0.002377 |
rf_results = pd.DataFrame(rf_grid.cv_results_)
rf_results.head(2)| mean_fit_time | std_fit_time | mean_score_time | std_score_time | param_bootstrap | param_max_depth | param_max_leaf_nodes | param_n_estimators | params | split0_test_score | split1_test_score | split2_test_score | split3_test_score | split4_test_score | mean_test_score | std_test_score | rank_test_score | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.557264 | 0.023031 | 0.025252 | 0.004624 | True | 4 | 2 | 250 | {'bootstrap': True, 'max_depth': 4, 'max_leaf_... | -4695.255873 | -4812.806394 | -4908.589487 | -5099.346146 | -5320.735759 | -4967.346732 | 220.760601 | 55 |
| 1 | 1.220450 | 0.056842 | 0.046306 | 0.003225 | True | 4 | 2 | 500 | {'bootstrap': True, 'max_depth': 4, 'max_leaf_... | -4692.579909 | -4812.292887 | -4926.218685 | -5100.746773 | -5321.148496 | -4970.597350 | 220.974408 | 58 |
fig, axes = plt.subplots(1,3,figsize=(14,5))
plt.subplots_adjust(wspace=0.2)
axes[0].plot(rf_results.param_max_leaf_nodes, (-rf_results.mean_test_score), 'o')
axes[0].set_xlabel('max leaf nodes')
axes[0].set_ylabel('MAE')
axes[1].plot(rf_results.param_n_estimators,(-rf_results.mean_test_score), 'o')
axes[1].set_xlabel('n estimators')
axes[1].set_ylabel('MAE');
axes[2].plot(rf_results.param_max_depth,(-rf_results.mean_test_score), 'o')
axes[2].set_xlabel(' max_depth')
axes[2].set_ylabel('MAE');2.3.2 Fine Grid Search
rf_params2 = {'n_estimators': [400, 700, 900],
'max_depth': [6, 8, 10],
'max_leaf_nodes': [250, 300]}
rf_grid2 = GridSearchCV(RandomForestRegressor(random_state = 4, bootstrap = True), rf_params2, n_jobs = -1, verbose = 1,
scoring = 'neg_root_mean_squared_error')
rf_grid2.fit(X_train_mars, y_train_mars)Fitting 5 folds for each of 18 candidates, totalling 90 fits
GridSearchCV(estimator=RandomForestRegressor(random_state=4), n_jobs=-1,
param_grid={'max_depth': [6, 8, 10], 'max_leaf_nodes': [250, 300],
'n_estimators': [400, 700, 900]},
scoring='neg_root_mean_squared_error', verbose=1)
print("Best RandomForest parameters:", rf_grid2.best_params_)
print("Best model train RMSE:", mean_squared_error(y_train_mars, rf_grid2.predict(X_train_mars), squared = False))
print("Best model test RMSE:", mean_squared_error(y_test_mars, rf_grid2.predict(X_test_mars), squared = False))
print("Best model R^2:", r2_score(y_test_mars, rf_grid2.predict(X_test_mars)))
rf_feats2 = pd.DataFrame(zip(rf_grid2.best_estimator_.feature_names_in_, rf_grid2.best_estimator_.feature_importances_)).reset_index()
rf_feats2.sort_values(by = rf_feats2.columns[2], ascending = False)Best RandomForest parameters: {'max_depth': 10, 'max_leaf_nodes': 300, 'n_estimators': 400}
Best model train RMSE: 1760.3116869136927
Best model test RMSE: 2411.105836013617
Best model R^2: 0.8343970360415414
| index | 0 | 1 | |
|---|---|---|---|
| 3 | 3 | year | 0.366251 |
| 0 | 0 | model_name | 0.264704 |
| 2 | 2 | power | 0.163420 |
| 1 | 1 | mileage | 0.134579 |
| 7 | 7 | savings_amount | 0.032096 |
| 4 | 4 | wheel_system_cat | 0.029813 |
| 5 | 5 | engine_cylinders_V6_Flex_Fuel_Vehicle | 0.005402 |
| 6 | 6 | fleet | 0.003734 |
Now we will continue with analysis on our selected features
base_rf = RandomForestRegressor(random_state = 4).fit(X_train, y_train)
print("Base RandomForest Test RMSE: ", mean_squared_error(y_test, base_rf.predict(X_test), squared = False))
print("Base RandomForest Train RMSE: ", mean_squared_error(y_train, base_rf.predict(X_train), squared = False))
print("Base RandomForest R^2: ", r2_score(y_test, base_rf.predict(X_test)))Base RandomForest Test RMSE: 2320.299016980909
Base RandomForest Train RMSE: 811.3938236091882
Base RandomForest R^2: 0.8494226618329384
base_rf_feats = pd.DataFrame(zip(base_rf.feature_names_in_, base_rf.feature_importances_)).reset_index()
base_rf_feats.sort_values(by = base_rf_feats.columns[2], ascending = False)| index | 0 | 1 | |
|---|---|---|---|
| 13 | 13 | trimId | 0.367903 |
| 8 | 8 | model_name | 0.206468 |
| 10 | 10 | power | 0.117995 |
| 7 | 7 | mileage | 0.109063 |
| 12 | 12 | torque | 0.028295 |
| 16 | 16 | year | 0.025966 |
| 11 | 11 | savings_amount | 0.024256 |
| 4 | 4 | height | 0.015664 |
| 6 | 6 | length | 0.013655 |
| 5 | 5 | highway_fuel_economy | 0.011459 |
| 3 | 3 | fuel_tank_volume | 0.011209 |
| 15 | 15 | width | 0.011103 |
| 21 | 21 | engine_cylinders_V6 | 0.010807 |
| 14 | 14 | wheelbase | 0.008779 |
| 18 | 18 | wheel_system_display_cat | 0.008285 |
| 17 | 17 | wheel_system_cat | 0.006988 |
| 0 | 0 | city_fuel_economy | 0.006192 |
| 1 | 1 | engine_displacement | 0.005792 |
| 2 | 2 | franchise_make | 0.004500 |
| 9 | 9 | owner_count | 0.003003 |
| 22 | 22 | transmission_display_8_Speed_Automatic | 0.001193 |
| 19 | 19 | body_type_Sedan | 0.000981 |
| 20 | 20 | engine_cylinders_I4 | 0.000442 |
#### Coarse GridSearchrf_params = {'n_estimators': [250, 500, 700],
'max_depth': [4, 6, 8],
'max_leaf_nodes': range(2, 300, 75),
'bootstrap': [True, False]}
rf_grid = GridSearchCV(RandomForestRegressor(random_state = 4), rf_params, n_jobs = -1, verbose = 1,
scoring = 'neg_root_mean_squared_error')
rf_grid.fit(X_train, y_train)
print("Best RandomForest parameters:", rf_grid.best_params_)
print("Best model train RMSE:", mean_squared_error(y_train, rf_grid.predict(X_train), squared = False))
print("Best model test RMSE:", mean_squared_error(y_test, rf_grid.predict(X_test), squared = False))
print("Best model R^2:", r2_score(y_test, rf_grid.predict(X_test)))
rf_feats = pd.DataFrame(zip(rf_grid.best_estimator_.feature_names_in_, rf_grid.best_estimator_.feature_importances_)).reset_index()
rf_feats.sort_values(by = rf_feats.columns[2], ascending = False)Fitting 5 folds for each of 72 candidates, totalling 360 fits
Best RandomForest parameters: {'bootstrap': True, 'max_depth': 8, 'max_leaf_nodes': 227, 'n_estimators': 700}
Best model train RMSE: 1798.8381156882924
Best model test RMSE: 2515.1964258302187
Best model R^2: 0.8230642798700507
| index | 0 | 1 | |
|---|---|---|---|
| 13 | 13 | trimId | 0.397924 |
| 8 | 8 | model_name | 0.227093 |
| 10 | 10 | power | 0.122868 |
| 7 | 7 | mileage | 0.102456 |
| 12 | 12 | torque | 0.028858 |
| 16 | 16 | year | 0.022507 |
| 4 | 4 | height | 0.011860 |
| 21 | 21 | engine_cylinders_V6 | 0.010883 |
| 6 | 6 | length | 0.010647 |
| 11 | 11 | savings_amount | 0.008886 |
| 5 | 5 | highway_fuel_economy | 0.008685 |
| 15 | 15 | width | 0.008171 |
| 3 | 3 | fuel_tank_volume | 0.007906 |
| 17 | 17 | wheel_system_cat | 0.007112 |
| 18 | 18 | wheel_system_display_cat | 0.007047 |
| 14 | 14 | wheelbase | 0.006170 |
| 1 | 1 | engine_displacement | 0.003463 |
| 0 | 0 | city_fuel_economy | 0.003062 |
| 9 | 9 | owner_count | 0.001666 |
| 2 | 2 | franchise_make | 0.001565 |
| 22 | 22 | transmission_display_8_Speed_Automatic | 0.000580 |
| 19 | 19 | body_type_Sedan | 0.000419 |
| 20 | 20 | engine_cylinders_I4 | 0.000169 |
fig, axes = plt.subplots(1,3,figsize=(14,5))
plt.subplots_adjust(wspace=0.2)
axes[0].plot(rf_results.param_max_leaf_nodes, (-rf_results.mean_test_score), 'o')
axes[0].set_xlabel('max leaf nodes')
axes[0].set_ylabel('MAE')
axes[1].plot(rf_results.param_n_estimators,(-rf_results.mean_test_score), 'o')
axes[1].set_xlabel('n estimators')
axes[1].set_ylabel('MAE');
axes[2].plot(rf_results.param_max_depth,(-rf_results.mean_test_score), 'o')
axes[2].set_xlabel('max_depth')
axes[2].set_ylabel('MAE');#### fine grid searchrf_params2 = {'n_estimators': [500, 700, 900],
'max_depth': [6, 8, 10],
'max_leaf_nodes': range(250, 350, 15)}
rf_grid2 = GridSearchCV(RandomForestRegressor(random_state = 4, bootstrap = True), rf_params2, n_jobs = -1, verbose = 1,
scoring = 'neg_root_mean_squared_error')
rf_grid2.fit(X_train, y_train)
print("Best RandomForest parameters:", rf_grid2.best_params_)
print("Best model train RMSE:", mean_squared_error(y_train, rf_grid2.predict(X_train), squared = False))
print("Best model test RMSE:", mean_squared_error(y_test, rf_grid2.predict(X_test), squared = False))
print("Best model R^2:", r2_score(y_test, rf_grid2.predict(X_test)))
rf_feats2 = pd.DataFrame(zip(rf_grid2.best_estimator_.feature_names_in_, rf_grid2.best_estimator_.feature_importances_)).reset_index()
rf_feats2.sort_values(by = rf_feats2.columns[2], ascending = False)Fitting 5 folds for each of 63 candidates, totalling 315 fits
Best RandomForest parameters: {'max_depth': 10, 'max_leaf_nodes': 340, 'n_estimators': 500}
Best model train RMSE: 1424.986609472238
Best model test RMSE: 2385.598110495618
Best model R^2: 0.8408281485903418
| index | 0 | 1 | |
|---|---|---|---|
| 13 | 13 | trimId | 0.384392 |
| 8 | 8 | model_name | 0.216342 |
| 10 | 10 | power | 0.119587 |
| 7 | 7 | mileage | 0.105961 |
| 12 | 12 | torque | 0.030705 |
| 16 | 16 | year | 0.022599 |
| 4 | 4 | height | 0.014340 |
| 11 | 11 | savings_amount | 0.014279 |
| 6 | 6 | length | 0.012538 |
| 21 | 21 | engine_cylinders_V6 | 0.010676 |
| 15 | 15 | width | 0.010466 |
| 5 | 5 | highway_fuel_economy | 0.009993 |
| 3 | 3 | fuel_tank_volume | 0.009782 |
| 14 | 14 | wheelbase | 0.007699 |
| 18 | 18 | wheel_system_display_cat | 0.007544 |
| 17 | 17 | wheel_system_cat | 0.007222 |
| 1 | 1 | engine_displacement | 0.004716 |
| 0 | 0 | city_fuel_economy | 0.004385 |
| 2 | 2 | franchise_make | 0.002616 |
| 9 | 9 | owner_count | 0.002076 |
| 22 | 22 | transmission_display_8_Speed_Automatic | 0.001105 |
| 19 | 19 | body_type_Sedan | 0.000651 |
| 20 | 20 | engine_cylinders_I4 | 0.000327 |
The best RMSE and R^2 resulted from the fine random forest grid search using our feature selection. This is because our feature selection was done using random forest.
Our best random forest RMSE and R2 are 2385.598110495618 and 0.8408281485903418 respectively.
2.4 CatBoost
By YuYan Zhang
import catboost as cb
from catboost import CatBoostRegressortrain_pool = cb.Pool(X_train,y_train)
test_pool = cb.Pool(X_test,y_test)cb1 = CatBoostRegressor(loss_function='RMSE')cb1.fit(train_pool)Learning rate set to 0.049132
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<catboost.core.CatBoostRegressor at 0x14b563f2520>
# train
print("Train RMSE:", mean_squared_error(y_train, cb1.predict(train_pool), squared = False))
print("Train R-squared:",r2_score(y_train, cb1.predict(train_pool)))Train RMSE: 1199.8271873207266
Train R-squared: 0.9597114208072007
# test
print("Test RMSE:", mean_squared_error(y_test, cb1.predict(test_pool), squared = False))
print("Test R-squared:",r2_score(y_test, cb1.predict(test_pool)))Test RMSE: 2055.989006548227
Test R-squared: 0.8817739227682351
feature_importance = cb1.feature_importances_
sorted_idx = np.argsort(feature_importance)
fig = plt.figure(figsize=(12, 6))
plt.barh(range(len(sorted_idx)), feature_importance[sorted_idx], align='center')
plt.yticks(range(len(sorted_idx)), np.array(X_test.columns)[sorted_idx])
plt.title('Feature Importance')Text(0.5, 1.0, 'Feature Importance')
cb2 = CatBoostRegressor()
cb_params = {'learning_rate':[0.01, 0.1],
'depth': [4, 7, 10],
'l2_leaf_reg': [2, 6, 10],
'random_strength': [0, 5, 10]
}
cb_search_result = cb2.grid_search(cb_params,
X=X_train,
y=y_train,
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bestTest = 2105.196343
bestIteration = 295
After visualizing the GridSearch results, we found that there continues to be the trend of way higher performance on train data, while test performance hovered around the same range. Therefore, we decided keep the untuned model.
2.5 XGBoost
By Emily Leibfritz
import time
from xgboost import XGBRegressorstart_time = time.time()
param_grid = {'max_depth': [8],
'learning_rate': [0.08,0.09,0.1],
'reg_lambda':[0.09],
'n_estimators':[510,515,520,530],
'gamma': [0.1,0.2],
'subsample': [0.75,0.8],
'colsample_bytree': [0.75,0.8]}
cv = KFold(n_splits=5,shuffle=True,random_state=4)
optimal_params = GridSearchCV(estimator=XGBRegressor(random_state=4),
param_grid = param_grid,
verbose = 1,
n_jobs=-1,
cv = cv)
optimal_params.fit(X_train,y_train)
print("Optimal parameter values =", optimal_params.best_params_)
print("Optimal cross validation R-squared = ",optimal_params.best_score_)
print("Time taken = ", round((time.time()-start_time)/60), " minutes")Fitting 5 folds for each of 96 candidates, totalling 480 fits
Optimal parameter values = {'colsample_bytree': 0.75, 'gamma': 0.2, 'learning_rate': 0.08, 'max_depth': 8, 'n_estimators': 510, 'reg_lambda': 0.09, 'subsample': 0.8}
Optimal cross validation R-squared = 0.8740821137070345
Time taken = 11 minutes
np.sqrt(mean_squared_error(optimal_params.best_estimator_.predict(X_test),y_test))2171.06738933653
XGBoost_model = XGBRegressor(random_state = 4, colsample_bytree =0.8, gamma = 0.2, learning_rate = 0.09,
max_depth = 8, n_estimators = 510, reg_lambda = 0.09, subsample = 0.8)
XGBoost_model.fit(X_train,y_train,eval_set = ([(X_test, y_test)]), early_stopping_rounds = 10)[0] validation_0-rmse:18343.24339
[1] validation_0-rmse:16730.82889
[2] validation_0-rmse:15276.12695
[3] validation_0-rmse:13960.30444
[4] validation_0-rmse:12756.05042
[5] validation_0-rmse:11671.49624
[6] validation_0-rmse:10691.13908
[7] validation_0-rmse:9793.15786
[8] validation_0-rmse:8972.68909
[9] validation_0-rmse:8239.99426
[10] validation_0-rmse:7570.86028
[11] validation_0-rmse:6955.64922
[12] validation_0-rmse:6410.91039
[13] validation_0-rmse:5921.50069
[14] validation_0-rmse:5486.91019
[15] validation_0-rmse:5093.51123
[16] validation_0-rmse:4743.50895
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[18] validation_0-rmse:4150.02788
[19] validation_0-rmse:3901.96935
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[91] validation_0-rmse:2160.65406
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[97] validation_0-rmse:2158.87579
[98] validation_0-rmse:2159.12650
XGBRegressor(base_score=None, booster=None, callbacks=None,
colsample_bylevel=None, colsample_bynode=None,
colsample_bytree=0.8, early_stopping_rounds=None,
enable_categorical=False, eval_metric=None, feature_types=None,
gamma=0.2, gpu_id=None, grow_policy=None, importance_type=None,
interaction_constraints=None, learning_rate=0.09, max_bin=None,
max_cat_threshold=None, max_cat_to_onehot=None,
max_delta_step=None, max_depth=8, max_leaves=None,
min_child_weight=None, missing=nan, monotone_constraints=None,
n_estimators=510, n_jobs=None, num_parallel_tree=None,
predictor=None, random_state=4, ...)
XGB_bagged = BaggingRegressor(base_estimator=XGBoost_model,
n_estimators=20,
bootstrap=False,
random_state=4,
n_jobs = -1).fit(X_train, y_train)np.sqrt(mean_squared_error(XGB_bagged.predict(X_test),y_test))2088.5007849998374
2.6 LGBM
By Emily Leibfritz
from lightgbm import LGBMRegressorLGMB_model=LGBMRegressor(random_state=4).fit(X_train,y_train)np.sqrt(mean_squared_error(LGMB_model.predict(X_test),y_test))2135.7915997846417
#K-fold cross validation to find optimal parameters for LightGBM regressor
start_time = time.time()
param_grid = {'max_depth': [4,5,6,7,8],
'num_leaves': [50,60,70,80,90,100,110,120,130,170,200],
'learning_rate': [0.05, 0.07,0.09,0.1,0.12,0.15],
'reg_lambda':[4,5,6,7,8,9,10],
'n_estimators':[800,900,1000,1100,1200,1300,1400,1500],
'reg_alpha': [1,2,3,4,5,6,7,8,9,10],
'subsample': [0.70, 0.75,0.8],
'colsample_bytree': [0.70,0.75,0.8, 0.9]}
cv = KFold(n_splits=5,shuffle=True,random_state=1)
optimal_params = RandomizedSearchCV(estimator=LGBMRegressor(random_state=1),
param_distributions = param_grid, n_iter = 200,
verbose = 1,
n_jobs=-1,
cv = cv)
optimal_params.fit(X_train,y_train)
print("Optimal parameter values =", optimal_params.best_params_)
print("Optimal cross validation R-squared = ",optimal_params.best_score_)
print("Time taken = ", round((time.time()-start_time)/60), " minutes")Fitting 5 folds for each of 200 candidates, totalling 1000 fits
Optimal parameter values = {'subsample': 0.75, 'reg_lambda': 10, 'reg_alpha': 10, 'num_leaves': 130, 'n_estimators': 1000, 'max_depth': 4, 'learning_rate': 0.05, 'colsample_bytree': 0.7}
Optimal cross validation R-squared = 0.8821169822021782
Time taken = 7 minutes
np.sqrt(mean_squared_error(optimal_params.best_estimator_.predict(X_test),y_test))2085.4135507703627
LGBM_bagged = BaggingRegressor(base_estimator=optimal_params.best_estimator_,
n_estimators=20,
bootstrap=False,
random_state=4,
n_jobs = -1).fit(X_train, y_train)np.sqrt(mean_squared_error(LGBM_bagged.predict(X_test),y_test))2078.6305836054316
3 Model Ensemble
Put code with comments. The comments should explain the code such that it can be easily understood. You may put text (in a markdown cell) before a large chunk of code to explain the overall purpose of the code, if it is not intuitive. Put the name of the person / persons who contributed to each code chunk / set of code chunks.
# models choosen for ensemble
bagged_tree = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=50, random_state=4,
oob_score=True,n_jobs=-1,bootstrap_features=True,bootstrap=True,
max_features=1.0,max_samples=1.0)
LGBM_bagged = BaggingRegressor(base_estimator=LGBMRegressor(subsample= 0.75, reg_lambda=10,
reg_alpha= 10, num_leaves=130,
n_estimators=1000, max_depth = 4,
learning_rate=0.05, colsample_bytree= 0.7),
n_estimators=20,
bootstrap=False,
random_state=4,
n_jobs = -1)
XGB_bagged = BaggingRegressor(base_estimator=XGBoost_model,
n_estimators=20,
bootstrap=False,
random_state=4,
n_jobs = -1)
catboost = CatBoostRegressor(loss_function='RMSE')3.1 Voting ensemble
en=VotingRegressor(estimators = [('bt',bagged_tree),('lgbm',LGBM_bagged),('xgb',XGB_bagged),('cb',catboost)])
en.fit(X_train,y_train)Learning rate set to 0.049132
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NameError: name 'Xtest' is not defined
print("Ensemble model RMSE = ", np.sqrt(mean_squared_error(en.predict(X_test),y_test)))Ensemble model RMSE = 2052.099261567051
3.2 Stacking ensemble(s)
stack_1 = StackingRegressor(estimators = [('bt',bagged_tree),('lgbm',LGBM_bagged),('xgb',XGB_bagged),('cb',catboost)],
final_estimator=LinearRegression(),
cv = KFold(n_splits = 5, shuffle = True, random_state=1))
stack_1.fit(X_train,y_train)Learning rate set to 0.049132
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StackingRegressor(cv=KFold(n_splits=5, random_state=1, shuffle=True),
estimators=[('bt',
BaggingRegressor(base_estimator=DecisionTreeRegressor(),
bootstrap_features=True,
n_estimators=50, n_jobs=-1,
oob_score=True,
random_state=4)),
('lgbm',
BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
learning_rate=0.05,
max_depth=4,
n_estimators=1000,
num...
max_delta_step=None,
max_depth=8,
max_leaves=None,
min_child_weight=None,
missing=nan,
monotone_constraints=None,
n_estimators=510,
n_jobs=None,
num_parallel_tree=None,
predictor=None,
random_state=4, ...),
bootstrap=False,
n_estimators=20, n_jobs=-1,
random_state=4)),
('cb',
<catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
final_estimator=LinearRegression())
print("Linear regression metamodel RMSE = ", np.sqrt(mean_squared_error(stack_1.predict(X_test),y_test)))Linear regression metamodel RMSE = 2033.444711920972
stack_2 = StackingRegressor(estimators = [('bt',bagged_tree),('lgbm',LGBM_bagged),('xgb',XGB_bagged),('cb',catboost)],
final_estimator=RandomForestRegressor(),
cv = KFold(n_splits = 5, shuffle = True, random_state=1))
stack_2.fit(X_train,y_train)Learning rate set to 0.049132
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StackingRegressor(cv=KFold(n_splits=5, random_state=1, shuffle=True),
estimators=[('bt',
BaggingRegressor(base_estimator=DecisionTreeRegressor(),
bootstrap_features=True,
n_estimators=50, n_jobs=-1,
oob_score=True,
random_state=4)),
('lgbm',
BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
learning_rate=0.05,
max_depth=4,
n_estimators=1000,
num...
max_delta_step=None,
max_depth=8,
max_leaves=None,
min_child_weight=None,
missing=nan,
monotone_constraints=None,
n_estimators=510,
n_jobs=None,
num_parallel_tree=None,
predictor=None,
random_state=4, ...),
bootstrap=False,
n_estimators=20, n_jobs=-1,
random_state=4)),
('cb',
<catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
final_estimator=RandomForestRegressor())
print("RF metamodel RMSE = ", np.sqrt(mean_squared_error(stack_2.predict(X_test),y_test)))RF metamodel RMSE = 2127.8992549473046
from sklearn.linear_model import LassoCVstack_3 = StackingRegressor(estimators = [('bt',bagged_tree),('lgbm',LGBM_bagged),('xgb',XGB_bagged),('cb',catboost)],
final_estimator=LassoCV(),
cv = KFold(n_splits = 5, shuffle = True, random_state=1))
stack_3.fit(X_train,y_train)Learning rate set to 0.049132
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StackingRegressor(cv=KFold(n_splits=5, random_state=1, shuffle=True),
estimators=[('bt',
BaggingRegressor(base_estimator=DecisionTreeRegressor(),
bootstrap_features=True,
n_estimators=50, n_jobs=-1,
oob_score=True,
random_state=4)),
('lgbm',
BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
learning_rate=0.05,
max_depth=4,
n_estimators=1000,
num...
max_cat_to_onehot=None,
max_delta_step=None,
max_depth=8,
max_leaves=None,
min_child_weight=None,
missing=nan,
monotone_constraints=None,
n_estimators=510,
n_jobs=None,
num_parallel_tree=None,
predictor=None,
random_state=4, ...),
bootstrap=False,
n_estimators=20, n_jobs=-1,
random_state=4)),
('cb',
<catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
final_estimator=LassoCV())
print("Lasso metamodel RMSE = ", np.sqrt(mean_squared_error(stack_3.predict(X_test),y_test)))Lasso metamodel RMSE = 2033.435450298843
3.3 Ensemble of ensembled models
voting_en=VotingRegressor(estimators = [('voting',en),('linear_stacking',stack_1)])
voting_en.fit(X_train,y_train)Learning rate set to 0.049132
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VotingRegressor(estimators=[('voting',
VotingRegressor(estimators=[('bt',
BaggingRegressor(base_estimator=DecisionTreeRegressor(),
bootstrap_features=True,
n_estimators=50,
n_jobs=-1,
oob_score=True,
random_state=4)),
('lgbm',
BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
learning_rate=0.05,
max_depth=4,
n_estimators=1000,
num_leaves=130,
r...
max_delta_step=None,
max_depth=8,
max_leaves=None,
min_child_weight=None,
missing=nan,
monotone_constraints=None,
n_estimators=510,
n_jobs=None,
num_parallel_tree=None,
predictor=None,
random_state=4, ...),
bootstrap=False,
n_estimators=20,
n_jobs=-1,
random_state=4)),
('cb',
<catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
final_estimator=RandomForestRegressor()))])
print("Voting Ensemble RMSE = ", np.sqrt(mean_squared_error(voting_en.predict(X_test),y_test)))Voting Ensemble RMSE = 2059.6950529918718
stacking_en=StackingRegressor(estimators = [('voting',en),('linear_stacking',stack_1)],
final_estimator=LinearRegression())
stacking_en.fit(X_train,y_train)Learning rate set to 0.049132
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StackingRegressor(estimators=[('voting',
VotingRegressor(estimators=[('bt',
BaggingRegressor(base_estimator=DecisionTreeRegressor(),
bootstrap_features=True,
n_estimators=50,
n_jobs=-1,
oob_score=True,
random_state=4)),
('lgbm',
BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
learning_rate=0.05,
max_depth=4,
n_estimators=1000,
num_leaves=130...
max_leaves=None,
min_child_weight=None,
missing=nan,
monotone_constraints=None,
n_estimators=510,
n_jobs=None,
num_parallel_tree=None,
predictor=None,
random_state=4, ...),
bootstrap=False,
n_estimators=20,
n_jobs=-1,
random_state=4)),
('cb',
<catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
final_estimator=RandomForestRegressor()))],
final_estimator=LinearRegression())
print("Stacking Ensemble RMSE = ", np.sqrt(mean_squared_error(stacking_en.predict(X_test),y_test)))Stacking Ensemble RMSE = 2040.0604592567781
Stacking with a Lasso and linear metamodel lead to the best results. Ensembling the ensembles did not lead to better performance.